{ "schema_version": 1, "upstream": { "repository": "https://github.com/leanprover-community/physlib", "revision": "9b583cb6ef6e9cc537c4d4b786b508de1abbcf42", "lean_toolchain": "leanprover/lean4:v4.35.0" }, "bindings": [ { "id": "absolute-temperature", "targets": [ { "kind": "node", "id": "Temperature", "uid": "63f9bd32d94d77e12afbb96f5b1b7a870de6b1f5805213b7f3c3ff227da301f0" } ], "relation": "conditional", "statement": "A nonnegative OMAI absolute-temperature value and Physlib Temperature round-trip exactly. The inverse-temperature formula and its numerical unit scaling are proved under explicit positivity and kB conventions.", "sources": [ { "module": "Physlib.Thermodynamics.Temperature.Basic", "name": "Temperature", "kind": "structure", "line": 32, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/Thermodynamics/Temperature/Basic.lean#L32" }, { "module": "Physlib.Thermodynamics.Temperature.Basic", "name": "Temperature.β", "kind": "def", "line": 59, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/Thermodynamics/Temperature/Basic.lean#L59" }, { "module": "Physlib.Thermodynamics.Temperature.Basic", "name": "Temperature.ofRealNonneg", "kind": "def", "line": 172, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/Thermodynamics/Temperature/Basic.lean#L172" }, { "module": "Physlib.Thermodynamics.Temperature.Basic", "name": "Temperature.beta_pos", "kind": "lemma", "line": 92, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/Thermodynamics/Temperature/Basic.lean#L92" } ], "conditions": [ "T>=0 for the carrier and value round trips; T>0 for beta positivity and unit-scaling results.", "For target numerical temperature a*T and energy b*E, use positive a,b and kB_target=(b/a)*kB_source; beta_target=beta_source/b. The upstream Boltzmann constant must satisfy that equality." ], "conventions": { "model_scope": "Absolute thermodynamic temperature.", "component_scope": "Scalar operating-point value.", "normalization_basis": "Absolute zero at zero.", "frequency_convention": "not applicable", "unit_embedding": "Kelvin or an explicitly consistent absolute-temperature scale; no Celsius offset.", "held_fixed": "not applicable", "admissibility": "Nonnegative carrier; strict positivity for beta-based laws.", "reference_energy": "not applicable", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "The caller must select and justify the absolute-temperature and energy units. This representation result does not verify a temperature-producing computation or a material model." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibContinuum", "declarations": [ "OMAI.PhyslibBridge.absoluteTemperature_toReal", "OMAI.PhyslibBridge.absoluteTemperature_roundtrip", "OMAI.PhyslibBridge.absoluteTemperature_beta_eq", "OMAI.PhyslibBridge.absoluteTemperature_beta_pos", "OMAI.PhyslibBridge.inverseTemperature_unit_scaling", "OMAI.PhyslibBridge.absoluteTemperature_beta_unit_scaling" ], "source_path": "lean/bridge/OMAIPhyslibContinuum.lean", "source_sha256": "e8bff3cb5d44cd78885d8c5f200d8761e8c568cd0d101fc0c2110f521bdca7ff", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "node", "id": "Temperature" } ] } }, { "id": "abstract-crystal-structure", "targets": [ { "kind": "node", "id": "Structure", "uid": "db1df71467f5a79440f849704ac20a33573ed4e270b5ebba59549272b2475239" } ], "relation": "conceptual", "statement": "Physlib Crystal represents periodicity by an additive-group action with a finite fundamental domain; OMAI Structure includes geometric/material content not yet identified.", "sources": [ { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal", "kind": "structure", "line": 65, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L65" } ], "conditions": [ "Choose the translation group and site equivalence matching the actual structure." ], "conventions": { "model_scope": "Abstract periodic crystal, not arbitrary nonperiodic/molecular structure.", "component_scope": "Sites and finite unit-cell representatives.", "normalization_basis": "Unit-cell correspondence must be explicit.", "frequency_convention": "not applicable", "unit_embedding": "Metric coordinates, cell vectors, and species are extra interpretation data.", "held_fixed": "not applicable", "admissibility": "not applicable", "reference_energy": "not applicable", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "Construct metric/species/cell identification; do not infer cell volume, atom counts, or electronic operators from abstract periodicity." ], "evidence": { "kind": "source_correspondence" } }, { "id": "analog-device-metrology", "relation": "local", "targets": [ { "kind": "node", "id": "ConductanceState", "uid": "325f09352a8872995fe6ab0a40b9df8780a778389fa0fe00ebcbd9ef7758f4a3" }, { "kind": "node", "id": "DotProductError", "uid": "1e6467a5b03397510e0d7f33403fd1cb42de5ea579e1f36b074d8dbbc620da20" }, { "kind": "node", "id": "ProgrammingPulseEnergy", "uid": "4dc358661b8a847db69d30a786b2b1475f1a291809bb34f55612d3203975be34" }, { "kind": "node", "id": "RetentionTime", "uid": "4280779257128d9b63d9180e7b7b03ab80beac66ee35480b178a09fa46b289f2" }, { "kind": "node", "id": "SetVoltage", "uid": "5b46cb4fc1bf84f1fa97e2b52a352694f5c98c856e9b9034763243994a1c2d21" }, { "kind": "operator", "id": "apply_conductance_drift", "uid": "f1e70e363faaf1f396092f32ad11a3807bdc9e0a4a11c95e8479a9d069e4f2ec", "formula_sha256": "4fd6bd7d557005282f8b1babd293cc0d95e6fc85b93deeb6c49956a491a8718f" }, { "kind": "operator", "id": "contract_device_conductance", "uid": "7f5e524af6d302713934d4a2bd5901abc64efdad9a5f8822bd19fe28b5b27cdf", "formula_sha256": "a57aff2c7d45172298a807999f389328df1fbd97817154a86ab9c138a420dd19" }, { "kind": "operator", "id": "dissipate_pulse_energy", "uid": "3714b2588a2c9400c356080644dab2fcdf778f5224235d6389519e648f41ab8a", "formula_sha256": "e9880765ed8745f0415055ed80e1770d0fcf29a8935ec292448b5594058bb8e7" }, { "kind": "operator", "id": "emit_thermionic_conductance", "uid": "58a959a412e7f92ec955cad34321812f5df453e9db8dcf4f8eb706f109bdf09a", "formula_sha256": "ae5ff88ffc8727046fd976c9097a5bb5745bb1908718b89aceb84914d38ed3da" }, { "kind": "operator", "id": "propagate_programming_error", "uid": "4517674f602a29f64ecc7140384fc486ee295233ce03ca7dc6bbec7eb00a91c2", "formula_sha256": "4977f8d380bdaa87a5000420f6c515cae3a2203ab4269d84301a6252b65c8917" }, { "kind": "operator", "id": "ramp_set_voltage", "uid": "9ab6278d7c06fcac5a8f9bcea7c65eb193c793c435822f24ca6762da34880dea", "formula_sha256": "dd1bf4e3e7338f03029e6d92efb5631aed804fe383d0cf857b8fb26703ce77a9" }, { "kind": "operator", "id": "zero_bias_retention", "uid": "10e6c7d5ab4087efdea29cf2219771cd55f19a87c50c19ca46756620116344e8", "formula_sha256": "0d49b118dcf2f3478766bb23218c8161e098df186770bcebe8779d0aa58aaf61" } ], "statement": "Every continuous positive drift law G_0 phi(t_d/t_{d,0}) that does not depend on the choice of reference time is the catalog's power law for some exponent (the multiplicative Cauchy equation). Conductances that follow the two-dimensional T^(3/2) injection law, inverted with the catalog's three-dimensional T^2 law from two temperatures T_1 < T_2, return Phi_Bn - k_B T_eff/2, where T_eff = log(T_2/T_1)/(1/T_1 - 1/T_2), the reciprocal of the logarithmic mean of 1/T_1 and 1/T_2, lies strictly between T_1 and T_2; this is the catalog's shift of about k_B T/2. An ensemble of devices with two symmetric sites exchanging at the catalog's retention rate 1/tau from the programming instant t = 0 on has a programmed fraction equal to 1/2 + (p(0) - 1/2) exp(-2t/tau) at every t >= 0, the only solution, relaxing to one half with time constant tau/2. Under a linear voltage ramp the expected hop count from zero bias is the leading term exp((V - V*)/b), b = k_B T/alpha_F, minus g_r k_B T/(alpha_F tau_0) with tau_0 = exp(E_m/(k_B T))/nu_0 the zero-bias lifetime of the unprogrammed state over the SET barrier; the first-hop law with survival exp(-exp((V - V*)/b)) has a density of total mass one, an integrable first moment, and mean V* - gamma b, the catalog's set voltage. The catalog's programming error exceeds c/sqrt(2 n_w) for every kernel of n_w nonzero weights and every window W > 1. For a conductance that rises or falls monotonically during a pulse, the dissipated energy lies between the catalog's pulse energies at the initial and the final conductance. For a channel of segments in series, the catalog's conductance with the length-averaged conductivity is an upper bound on the true conductance.", "sources": [], "conditions": [ "Energies Phi_Bn and E_m are in joules, T in kelvin, k_B and q_e exact SI; P_th in A/K^2 and G in siemens. An energy in eV entered as joules inside exp(-E/(k_B T)) gives an exponentially wrong result, not a constant factor.", "Drift: a single device and a single programmed level with one fixed exponent nu, isothermal storage, positive times inside the time range where nu was fitted, and 0 < t_{d,0}. An array mean over devices with a spread of nu is not a power law (a Gaussian nu of mean mu and width sigma gives d ln/d ln t = -mu + sigma^2 ln(t/t_0)).", "Thermionic: one contact, thermionic emission over a laterally homogeneous barrier without field emission, small signal |V| much smaller than k_B T/q_e, and an ideality factor independent of temperature; the catalog's G is the zero-bias slope for ideality factor 1. Phi_Bn is the effective zero-bias barrier. The zero-bias Richardson plot of this conductance is ln(G/T) against 1/T, not ln(G/T^2). A Gaussian spread sigma of barrier heights (Werner and Guttler) makes a two-temperature inversion return Phi_mean - sigma^2 (1/T_1 + 1/T_2)/(2 k_B). The catalog stores this small-signal slope in ConductanceState, whose meaning elsewhere is the chord conductance.", "Retention: one back-hop path at the Arrhenius rate nu_0 exp(-E_m/(k_B T)), or M equivalent paths. The relaxation to one half holds for symmetric sites, where the device re-sets as often as it decays. At one temperature E_m and nu_0 cannot be separated. A datasheet retention is a time to failure at a fail fraction, not this mean; a spread of barrier heights makes the survival non-exponential. The rate equation is assumed for t >= 0 only; nothing is assumed about the fraction before programming.", "Ramp: a linear ramp with dwell g_r per volt (1/ramp rate, or t_p/dV for a staircase), alpha_F > 0, a barrier falling linearly with V, one hop switching the device, the device at temperature T. The mean set voltage uses the leading hop count and neglects hops at zero and negative bias, whose expected number g_r k_B T/(alpha_F tau_0) must be much smaller than one; tau_0 = exp(E_m,set/(k_B T))/nu_0 is the zero-bias lifetime of the unprogrammed state over the SET-direction barrier, not the catalog's RetentionTime of the programmed state, and the two agree only for symmetric sites.", "Programming error: c > 0, W > 1, w_max > 0, n_w the count of nonzero weights, S_1 = sum |w_i| and S_2 = sum w_i^2. The Lean shows that the catalog's closed form equals the per-cell sum c sqrt(sum_i (w_max/(W - 1) + |w_i|)^2/2)/S_1 of the catalog description and bounds it; it does not derive that expression from the stochastic cell model. Read noise, drift, IR drop, sneak paths and correlated errors are not included.", "Pulse energy: V_p is the voltage across the same two terminals between which G is defined, with no series resistor, selector or compliance in that path; G is the chord conductance I(V_p)/V_p and changes monotonically during the pulse (rising for SET, falling for RESET). A gate pulse is not covered.", "Device conductance: current flows along the channel through segments of positive length and conductivity with one cross-section A; the catalog's sigma is then the length-averaged conductivity and the catalog conductance an upper bound, equal for a uniform channel." ], "conventions": { "model_scope": "Closed-form device models: power-law drift, thermionic emission over a Schottky barrier, Arrhenius hopping for retention and ramped set, a per-cell programming-error model, Joule dissipation and series conduction. Physlib has no device model, so these are local theorems about the OMAI formulas over Mathlib reals; no Physlib declaration is identified.", "component_scope": "One device, one contact or one cell array as each operator describes; scalar quantities.", "normalization_basis": "Conductances per device; the programming error is relative to the ideal dot product; hop counts are expected numbers per device.", "frequency_convention": "nu_0 is an attempt frequency in hops per unit time (THz in the catalog), not an angular frequency.", "unit_embedding": "The catalog stores times in ps and frequencies in THz; with joules, siemens and volts this embedding is not coherent: V_p^2 G t_p in V^2 S ps is a picojoule. The theorems hold in any coherent embedding; convert to one before comparing values.", "held_fixed": "Temperature during a measurement; for the two-temperature inversions the barrier and prefactor are held fixed between the two temperatures; the drift exponent within its fitted range.", "admissibility": "Positive temperatures, times, attempt frequency, alpha_F, g_r and P_th; W > 1 and c > 0 for the programming error; monotone conductance during a pulse; positive segment lengths and conductivities.", "reference_energy": "Barriers are measured from the state that is left: the programmed state for retention, the unprogrammed state for the ramp.", "qpoint_weighting": "Not applicable." }, "remaining_obligations": [ "Measured or fitted parameters (nu, P_th, Phi_Bn, E_m, nu_0, alpha_F, c) and the measurements behind them are not verified.", "Drift: a spread and state dependence of nu across devices and levels, nu(T) with thermally accelerated time, non-isothermal storage and 1/f read noise are not covered.", "Thermionic: field and thermionic-field emission, the image-force temperature dependence of the barrier, barrier inhomogeneity and a temperature-dependent ideality factor are not covered.", "Retention: the minimum of M independent exponential exit times and failure-fraction (datasheet) retention are not proved; only the mean and the symmetric two-site relaxation are.", "The Gumbel standard deviation pi k_B T/(sqrt(6) alpha_F), the N-hop generalization (k_B T/alpha_F)(psi(N) + gamma), device-to-device barrier spread, staircase discretization and Joule heating are not proved.", "The programming-error model's derivation from the stochastic cell model is not proved; read noise, drift, IR drop, sneak paths, quantization and correlated errors are excluded.", "Capacitive charging energy, drive-electronics energy, contact resistance and short-channel (Landauer) transport are not covered.", "contract_conductance_window and contract_state_coefficient_of_variation are definitions with nothing non-obvious to prove and are not bound; schottky_mott_barrier is unbound because its validity cannot be established algebraically." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIDeviceMetrology", "declarations": [ "OMAI.PhyslibBridge.DeviceMetrology.drift_law_unique", "OMAI.PhyslibBridge.DeviceMetrology.thermionic_wrongLaw_barrier", "OMAI.PhyslibBridge.DeviceMetrology.retention_symmetric_relaxation", "OMAI.PhyslibBridge.DeviceMetrology.ramp_expectedHops", "OMAI.PhyslibBridge.DeviceMetrology.ramp_meanSetVoltage", "OMAI.PhyslibBridge.DeviceMetrology.programmingError_gt", "OMAI.PhyslibBridge.DeviceMetrology.pulseEnergy_bracket", "OMAI.PhyslibBridge.DeviceMetrology.deviceConductance_series_le" ], "source_path": "lean/bridge/OMAIDeviceMetrology.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "apply_conductance_drift" }, { "kind": "operator", "id": "contract_device_conductance" }, { "kind": "operator", "id": "dissipate_pulse_energy" }, { "kind": "operator", "id": "emit_thermionic_conductance" }, { "kind": "operator", "id": "propagate_programming_error" }, { "kind": "operator", "id": "ramp_set_voltage" }, { "kind": "operator", "id": "zero_bias_retention" }, { "kind": "node", "id": "ConductanceState" }, { "kind": "node", "id": "DotProductError" }, { "kind": "node", "id": "ProgrammingPulseEnergy" }, { "kind": "node", "id": "RetentionTime" }, { "kind": "node", "id": "SetVoltage" } ], "source_sha256": "d03439a60245074d824ec390bbe303cfb149d667db2281555428f6d385a36e91" } }, { "id": "arrhenius-ionic-transport", "relation": "local", "targets": [ { "kind": "node", "id": "ActivationEnergy", "uid": "5d35f8f6975629291c51d432d3b137fb5bd92165706d37dbab71366a4714b71d" }, { "kind": "node", "id": "ElectricalConductivity[carrier=ionic]", "uid": "3f5753096d30cc6741fe9c6fd5e247e9afbb87b94f5844ec233f9243345d3712" }, { "kind": "operator", "id": "compute_ionic_conductivity", "uid": "d4303ca638ffb5ca529f8c681909d728647149d1b79fd02ec8a18c9270afc05c", "formula_sha256": "8beec3f9dffff073f2679da513e2faea9a31883e2e816ad3815c04d18cdc6af5" }, { "kind": "operator", "id": "fit_arrhenius", "uid": "1038d2249c062baff931bbc06138496d90157bb4365334cae8c5092866e9b6a4", "formula_sha256": "30b75223fc1faba9136c4731d024ff4a258a6bb2ff8dcb7a170ffc25f9b633d5" } ], "statement": "The catalog's weighted Arrhenius fit is the weighted least-squares line of log D against 1/T: its slope minimizes the weighted sum of squared residuals over all lines. On data that follow the catalog's Arrhenius law exactly, it returns E_a = -k_B times the slope, for every choice of positive weights and at least two distinct temperatures; this is an exact-data identity, not a statement about noisy data. Fitting the catalog's Nernst-Einstein conductivity sigma itself, rather than sigma T, in the same way returns the apparent activation energy E_a - k_B T_eff with T_eff inside the temperature range of the data, for every choice of positive weights, when the diffusivity is Arrhenius with barrier E_a and the carrier density does not depend on temperature: sigma T, not sigma, carries the diffusion barrier.", "sources": [], "conditions": [ "E_a is an energy per particle in joules, k_B exact SI, T in kelvin at every data point; D_0 > 0. The regression is of log D against 1/T with positive weights and at least two distinct temperatures.", "Nernst-Einstein with the tracer diffusivity and Haven ratio H_R = 1; the carrier density n_c, the charge number z (nonzero), the elementary charge e_c and the Arrhenius parameters do not depend on temperature.", "The Einstein relation behind the input diffusivity needs d to match the number of MSD components summed and the diffusive regime of the MSD." ], "conventions": { "model_scope": "Arrhenius thermally activated diffusion and the Nernst-Einstein relation. Physlib has no solid-state diffusion model, so these are local theorems about the OMAI formulas over Mathlib reals; no Physlib declaration is identified.", "component_scope": "One mobile species with a scalar diffusivity and conductivity.", "normalization_basis": "Activation energies per particle; carrier density per unit volume.", "frequency_convention": "Not applicable.", "unit_embedding": "Coherent SI: D in m^2/s, E_a in J, k_B in J/K, T in K, n_c in 1/m^3, e_c in C, sigma in S/m.", "held_fixed": "Carrier density, charge number and Arrhenius parameters across the temperature range.", "admissibility": "T > 0 at every data point, positive weights, D_0 > 0, n_c > 0, z and e_c nonzero.", "reference_energy": "Not applicable.", "qpoint_weighting": "Not applicable." }, "remaining_obligations": [ "Molecular-dynamics or experimental producers of the diffusivity and noise in fitted data are not verified; the exactness theorem covers exact Arrhenius data only.", "Non-Arrhenius curvature, correlated conduction (Haven ratio different from 1) and temperature-dependent carrier densities are not covered.", "contract_diffusivity (D = slope/(2d)) is a definition with nothing non-obvious to prove and is not bound; compute_carrier_density and compute_configurational_energy are external selectors with nothing algebraic to prove." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIMaterialsTransport", "declarations": [ "OMAI.PhyslibBridge.MaterialsTransport.wlsSlope_minimizes", "OMAI.PhyslibBridge.MaterialsTransport.arrhenius_wls_recovers_activation", "OMAI.PhyslibBridge.MaterialsTransport.ionic_wls_apparent_activation" ], "source_path": "lean/bridge/OMAIMaterialsTransport.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_ionic_conductivity" }, { "kind": "operator", "id": "fit_arrhenius" }, { "kind": "node", "id": "ActivationEnergy" }, { "kind": "node", "id": "ElectricalConductivity[carrier=ionic]" } ], "source_sha256": "2aa396f9e92d7b2a52b7d8cceb05c7834dee771777c23c47094dd2b2ed82d379" } }, { "id": "boltzmann-electronic-transport", "targets": [ { "kind": "node", "id": "ElectricalConductivity[carrier=electronic]", "uid": "c4a81f649b3964aed4d4b3c48dfec4f252958c9560332286e2105b175794144b" }, { "kind": "node", "id": "ElectronicThermalConductivity", "uid": "960b38ae04ecc9fbaacb02d9ae832f15e84b8cac0563e7f6a62ddf32c61d725c" }, { "kind": "node", "id": "SeebeckCoefficient", "uid": "80778c5d6cb9f0d1dd23d5c6cd6059acea161892532ce7c581c7477257debf02" }, { "kind": "operator", "id": "compute_electronic_conductivity", "uid": "9b20bb39ed27457a9f0c5655ab1001b0abdcf86c4ab4883c87e8a54847e0ab72", "formula_sha256": "2de7ab9fcfdde2fac6c5e3ab0e88316483e11a22ae90687e08aa8897908580e0" }, { "kind": "operator", "id": "compute_electronic_thermal_conductivity", "uid": "b24757dc6349ac2bcf8429ab264385f76d6895fc52baea4eb380f7d7e063ca11", "formula_sha256": "7ffb8ea9b51164875b7f3eaa724d017c8c846b3fd22f4da0befa6169596c9586" }, { "kind": "operator", "id": "compute_seebeck", "uid": "f34ee134bfed051ae79dfece5f9ff45df4c2c202e784ccf6c3bf50035709d208", "formula_sha256": "75307432c842b2c58ffc978745dd0992e89936a44fab7dc40bb06a24a887b939" } ], "relation": "conditional", "statement": "The producers are interpreted as BoltzTraP2's Onsager coefficients on a finite energy grid: with weights w_b >= 0 (the diagonal transport distribution times the Fermi window -df/dE times the bin width), energies E_b and L_k = sum_b w_b (E_b - mu)^k, sigma = e^2 L_0, S = -(L_1/L_0)/(e T) and kappa_e = (L_2 - L_1^2/L_0)/T. At T > 0 the electronic thermal conductivity is nonnegative and vanishes exactly when every state with nonzero weight has one energy. For a window with L_0 > 0, e != 0, T > 0 and any lattice conductivity kappa_l > 0, these coefficients are the S, sigma, kappa_l and kappa_e fields of a Physlib ThermoelectricMaterial. When L_0 L_2 - L_1^2 > 0 and L_1 != 0, every Physlib material carrying these S, sigma and kappa_e has figure of merit zT = sigma S^2 T/(kappa_l + kappa_e) strictly below L_1^2/(L_0 L_2 - L_1^2), the squared mean over the variance of E - mu in the transport window (the Mahan-Sofo bound), and materials with small enough kappa_l come within any eps > 0 of it. For the continuum kinetic integrals K_k = int Sigma_0 (E - mu)^k (-df/dE) dE of an energy-independent transport distribution Sigma_0 at k_B T > 0, the Fermi-window moments give K_0 = Sigma_0, K_1 = 0 and K_2 = Sigma_0 (k_B T)^2 pi^2/3, so the same three formulas give S = 0 and kappa_e = (pi^2/3)(k_B/e)^2 sigma T, the Sommerfeld Lorenz number.", "sources": [ { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial", "kind": "structure", "line": 102, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L102" }, { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.figureOfMerit", "kind": "def", "line": 158, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L158" }, { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.figureOfMerit_eq", "kind": "lemma", "line": 169, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L169" } ], "conditions": [ "compute_electronic_conductivity, compute_seebeck and compute_electronic_thermal_conductivity are opaque Boltzmann-transport outputs in the catalog. The interpreted definitions are BoltzTraP2's Onsager coefficients evaluated on a finite energy grid, L_k = sum_b w_b (E_b - mu)^k with w_b = Sigma(E_b) (-df/dE)(E_b) Delta E >= 0 and Sigma the diagonal transport distribution sum_k v_k^2 tau_k delta(E - E_k)/(N_k V). The relaxation time enters only through w, so amset's state-resolved (band and k-point) rates and a constant relaxation time are both admitted; one diagonal Cartesian component (or one principal axis) at a time, since off-diagonal weights v_a v_b are not sign definite. The catalog's bte_mrta scheme is amset's momentum-relaxation-time solution, whose state weights v_k^2 tau_k are nonnegative; a self-consistent iterative BTE solution would have no state-resolved nonnegative weights and is outside these hypotheses.", "e > 0 is the elementary charge, so S = -(L_1/L_0)/(e T) is negative when the transport window lies above mu (n-type). A code that writes S = (L_1/L_0)/(q T) with the signed carrier charge q = -e gives the same number. kappa_e is the zero-current electronic thermal conductivity, with -L_1^2/L_0 the Peltier correction; the zero-field L_2/T is a different quantity.", "The Physlib material needs L_0 > 0 (some state with nonzero weight), e != 0 and T > 0 for sigma > 0, and an independently supplied lattice conductivity kappa_l > 0; kappa_e >= 0 is proved, not assumed. The figure of merit is Physlib's zT = sigma S^2 T/(kappa_l + kappa_e).", "The Mahan-Sofo bound needs L_0 L_2 - L_1^2 > 0, which for L_0 > 0 means that the states with nonzero weight do not all share one energy, and L_1 != 0; it is strict for every kappa_l > 0 and sharp as kappa_l -> 0. It bounds the figure of merit by the shape of the transport window alone and says nothing about which window is attainable.", "The Sommerfeld statements are about the continuum kinetic integrals of an energy-independent transport distribution Sigma_0 at k_B T > 0 (the degenerate limit with a flat Sigma), not about the finite-grid definitions; no condition on Sigma_0 is needed, since K_1 = 0 makes the Peltier correction K_1^2/K_0 vanish; the Lorenz number (pi^2/3)(k_B/e)^2 is the Sommerfeld value 2.44e-8 W Ohm/K^2 in SI.", "Energies in joules, T in kelvin, e in coulombs; sigma in S/m, S in V/K and kappa_e in W/(m K) when w carries S/(m C^2). BoltzTraP2's hartree energies and amset's unit choices (the catalog description flags its kappa unit as unconfirmed) are conversions outside the theorems." ], "conventions": { "model_scope": "Linearized Boltzmann transport in the relaxation-time form of BoltzTraP2 and amset, interpreted as Onsager moments of a nonnegative transport window on a finite energy grid; anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit.", "component_scope": "One diagonal Cartesian component of sigma, S and kappa_e at a time (nonnegative weights); the full tensors and their off-diagonal components are not covered.", "normalization_basis": "Weights per energy bin, transport distribution per unit volume and per k-point count as BoltzTraP2 defines it; L_0 carries the e^-2 of sigma.", "frequency_convention": "Not applicable; no phonon frequency enters these statements. The phonon-frequency input of the catalog producers acts only through the scattering rates inside w.", "unit_embedding": "Coherent SI: energies and mu in J, T in K, e in C, sigma in S/m, S in V/K, kappa_e in W/(m K). Exact-real statements; floating-point evaluation is not certified.", "held_fixed": "Chemical potential, temperature, the transport window w and the lattice conductivity are fixed inputs; the Mahan-Sofo limit varies kappa_l alone.", "admissibility": "w_b >= 0 for every grid point; T > 0; L_0 > 0 and e != 0 for the Physlib material; L_0 L_2 - L_1^2 > 0 and L_1 != 0 for the bound; k_B T > 0 for the Sommerfeld moments; e != 0 and T > 0 for the Lorenz-number form, with no condition on Sigma_0.", "reference_energy": "Energies are measured from the chemical potential mu inside L_k; a common shift of E_b and mu leaves every coefficient unchanged.", "qpoint_weighting": "The k-point sum and its weights are inside the transport distribution Sigma(E); the theorems see only the resulting energy-grid weights w_b." }, "remaining_obligations": [ "The band structure, its BoltzTraP2 interpolation, the energy grid, the Fermi-window discretization and amset's scattering rates (ADP, IMP, PIE, POP) that build w are not verified.", "Off-diagonal tensor components, Hall and magnetic-field coefficients, bipolar transport with two carrier types and the iterative (beyond relaxation-time) BTE solution are not covered.", "The Sommerfeld moments are proved for a flat transport distribution; the Sommerfeld expansion of a smooth Sigma(E) and its corrections are not.", "Whether the Mahan-Sofo bound is approached by any real band structure is not addressed; the sharpness statement varies the lattice conductivity only." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibElectronicTransport", "declarations": [ "OMAI.PhyslibBridge.ElectronicTransport.boltzmannKappaE_nonneg_and_eq_zero_iff", "OMAI.PhyslibBridge.ElectronicTransport.exists_boltzmann_thermoelectricMaterial", "OMAI.PhyslibBridge.ElectronicTransport.figureOfMerit_lt_mahanSofo", "OMAI.PhyslibBridge.Sommerfeld.sommerfeld_moments", "OMAI.PhyslibBridge.Sommerfeld.wiedemannFranz_constantTransport" ], "source_path": "lean/bridge/OMAIPhyslibElectronicTransport.lean", "build_record": "lean/bridge/build-evidence.json", "source_sha256": "2609b6b410cb84b120f782adf88b7204e13bcef14ddee5adbce92f5c89805f45" } }, { "id": "calphad-binary-equilibrium", "relation": "local", "targets": [ { "kind": "node", "id": "CalphadMolarEntropy", "uid": "b11c4ec0ddbfa4483cc0a65d6a8fb2cefa9687b9a86ebd11bfb968d4253ec022" }, { "kind": "node", "id": "ChemicalPotential", "uid": "e707090e106994d3718f0afe5a20f2e60037f543334fa79b4a00360b7962d527" }, { "kind": "node", "id": "MolarEnthalpy", "uid": "600d0860290d3326202768db7249339c34ac675deb2ee8528a92207d3878b00c" }, { "kind": "node", "id": "MolarGibbsEnergy", "uid": "b048bc7569d568e2201557761eb758817674dd3a38b6f2379c5474f4cf91c66f" }, { "kind": "node", "id": "PhaseFraction", "uid": "7b55b11531084dcb2d5fd95100d62b2d68b787596e7c9712e06c011817ab5b8c" }, { "kind": "node", "id": "TransitionTemperature", "uid": "8b9dd2a79db11c01e15d37f64a7bd0d9c95f09ea080a67f7912d9b8aabd7b132" }, { "kind": "operator", "id": "compute_calphad_entropy", "uid": "c1a786957352c7a02efab90d3815c83156bbae53976932e51dec1a9f0df62f2b", "formula_sha256": "841219f58ee33183c89a9fcd83b3c4a34c426c7be1473d45431ccd81b756c7cf" }, { "kind": "operator", "id": "compute_chemical_potentials", "uid": "6e99a335cfb6b775c858856c9f81be42bf54bb931975aa16e4b13eedfa874b61", "formula_sha256": "a95d84ace50d2262f60cc554ca21e03163140ec272d406f065bd39d9d6610fcb" }, { "kind": "operator", "id": "compute_molar_enthalpy", "uid": "901873fa2497cc75e75079357e7125527526bf2590507835f5cbeae3e5fd2868", "formula_sha256": "559358c8898996d1b25239cf7891b8cad1297e6964a661b04055f8cf53e78812" }, { "kind": "operator", "id": "compute_phase_fractions", "uid": "d5a6a4d1c789d427aecbc40998ec2b7d32240c2850ac5d9ed226e268b6ed09f8", "formula_sha256": "cdec0b486b6a7f7bbcc6f823a398f698ab01889104b0374971dd06ab7bf1397a" }, { "kind": "operator", "id": "compute_transition_temperature", "uid": "6f9aee9e741ba70cb9ee59098485a719edfaa20a864e0a5122ae7a225a875363", "formula_sha256": "dff18b12ad9924ead14269682911a8017a719684f240433725bdcdf9b4789c80" }, { "kind": "operator", "id": "solve_equilibrium", "uid": "8e00f3c45c66888481ee6a2004e915534c06dfbc94ef59c7c761b85a1a7365a8", "formula_sha256": "e462f6823acba4718605093dd6a9c2f5b65d76e52a2f0f936f34b1d30f06d951" } ], "statement": "For a binary system the interpreted equilibrium molar Gibbs energy, the least energy of any finite mixture of phases with the overall composition, is the convex envelope of the phase Gibbs curves on [0, 1]: it is convex, lies below every phase curve and lies above every convex function below them; the phase set need not be finite and the infimum need not be attained. If every phase's Gibbs function is concave in temperature on a temperature interval, the equilibrium Gibbs energy at any fixed composition is concave in temperature there, so on the interior of that interval the equilibrium entropy S = -dG/dT and, at nonnegative temperatures, the equilibrium enthalpy H = G - T dG/dT (right derivatives) are nondecreasing through every phase transition: the equilibrium heat capacity, latent heats included, is nonnegative. At a two-phase local extremum with interior compositions both differentiable phase curves share the common tangent, so both phases have the same chemical potentials, and where the equilibrium envelope follows that two-phase chord its tangent intercepts, the catalog's chemical potentials, are the common-tangent intercepts. Mass balance fixes the two phase fractions to the lever rule, both in [0, 1] exactly when the overall composition lies between the phase compositions. For a unary system or a congruent transformation, a Gibbs-energy difference that changes sign with a positive entropy difference has exactly one transition temperature, with phase alpha stable below it and phase beta above.", "sources": [], "conditions": [ "Every producer except contract_gibbs_hts is an opaque solver call in the catalog. The interpretations read molar Gibbs functions of the assessed phases per mole of atoms at fixed pressure with the SER reference, as pycalphad's GM, HM, SM, MU and NP; they are not the harmonic quantities per mole of primitive cells of the phonon bridge, and Gibbs is not Helmholtz.", "The equilibrium and chemical-potential results are for a binary system, composition the mole fraction x of the second component in [0, 1], at the temperature and pressure built into the phase functions. The phase Gibbs curves are bounded below on [0, 1]; ideal-mixing terms x ln x are bounded there.", "compute_molar_enthalpy and compute_calphad_entropy use the right temperature derivative of the equilibrium Gibbs energy at fixed composition, the values of the state stable just above T, equal to G - T G' and -G' wherever G is differentiable. The monotonicity results need every phase Gibbs function concave in temperature (nonnegative phase heat capacities) on the interval and hold on its interior; assessed functions extrapolated outside their valid range can have negative heat capacity. Within a two-phase field -T d2G_eq/dT2 is the apparent heat capacity, latent heat of the moving phase boundary included, not pycalphad's phase-local value.", "compute_chemical_potentials returns the tangent intercepts mu_A = G - x G' and mu_B = G + (1 - x) G' of the equilibrium envelope G. The common-tangent theorem assumes a local extremum of the two-phase mixture energy in each phase composition (the proof uses only the vanishing derivative) with x1 < x < x2 and both phase curves differentiable there, which excludes line compounds; it does not establish that the global minimum is two-phase. The envelope-intercept theorem assumes that the equilibrium envelope agrees with the two-phase chord on a neighbourhood of x.", "compute_transition_temperature is covered only for a unary system or a congruent (partitionless) transformation between two phases of one composition, bracketed in [T1, T2] by a sign change of G_beta - G_alpha with S_beta > S_alpha inside. For an alloy the temperature of equal phase Gibbs energies at one composition is the T0 temperature, not the catalog's boundary locus (liquidus, solidus or solvus, where a phase fraction reaches 0 or 1): an ideal lens at x = 0.5 has T0 = 1250.0 K, solidus 1219.5 K and liquidus 1279.1 K (T_A = 1000 K, T_B = 1500 K, equal fusion entropies of 10 J/(mol K), Delta C_p = 0), with a two-phase equilibrium across that window. Continuous transitions (order-disorder, magnetic) are excluded.", "contract_gibbs_hts, G = H - T S, is a definition with nothing to prove beyond these interpretations and is not a proof subject." ], "conventions": { "model_scope": "Equilibrium thermodynamics of assessed CALPHAD phase models. Physlib has no model of Gibbs minimization over assessed phases, so these are local theorems over Mathlib reals that identify no Physlib declaration.", "component_scope": "Binary systems for the equilibrium, chemical-potential and phase-fraction results; any fixed composition for the temperature monotonicity; one composition and two phases of a unary or congruent transformation for the transition temperature.", "normalization_basis": "Per mole of atoms (pycalphad v.N = 1), not per mole of formula units or primitive cells.", "frequency_convention": "Not applicable; no frequencies enter these formulas.", "unit_embedding": "Molar energies in J/mol, entropies in J/(mol K), temperatures in kelvin, compositions as mole fractions in [0, 1], not percentages.", "held_fixed": "Pressure throughout; composition for the temperature derivatives; temperature for the composition derivatives and the equilibrium minimization.", "admissibility": "Phase Gibbs curves bounded below on [0, 1] (at each temperature of the interval for the temperature results); phase Gibbs functions concave in temperature for the monotonicity results and T >= 0 for the enthalpy; x1 < x2 for the lever rule and x1 < x < x2 for the common tangent; T1 < T2 with a sign change and positive entropy difference for the transition.", "reference_energy": "The SER reference of the assessed database. A reference shift linear in composition shifts G, mu_A and mu_B consistently; its effect on phase fractions and transition temperatures is not proved here (see the remaining obligations).", "qpoint_weighting": "Not applicable; no Brillouin-zone sums enter these formulas." }, "remaining_obligations": [ "The fidelity of the assessed database to experiment and the solver's global minimization (pycalphad's grid sampling and refinement) are not verified; the theorems hold for the interpreted definitions.", "Multicomponent systems, sublattice site-fraction minimization inside a phase, magnetic and ordering contributions as separate terms, and pressure dependence are not covered.", "That the global minimum is attained by at most two phases in a binary system (Caratheodory) and that a two-phase minimum exists between the common-tangent compositions are not proved.", "That the equilibrium envelope agrees with the two-phase chord near x is not derived from the envelope definition (it needs the two-phase attainment above); given that agreement, the envelope's tangent intercepts equal the common-tangent intercepts.", "The catalog's boundary locus for alloys (liquidus, solidus, solvus and invariant points in composition-temperature space, extracted from a swept phase-fraction grid) is not verified; only the unary or congruent transition is.", "Invariance of phase fractions and transition temperatures under a reference shift linear in composition is not proved." ], "evidence": { "kind": "compiled_bridge", "module": "OMAICalphad", "declarations": [ "OMAI.PhyslibBridge.Calphad.equilibriumGibbs_isConvexEnvelope", "OMAI.PhyslibBridge.Calphad.equilibriumGibbsT_concaveOn", "OMAI.PhyslibBridge.Calphad.equilibrium_entropy_monotoneOn", "OMAI.PhyslibBridge.Calphad.equilibrium_enthalpy_monotoneOn", "OMAI.PhyslibBridge.Calphad.commonTangent_of_isLocalExtr", "OMAI.PhyslibBridge.Calphad.deriv_equilibriumGibbs_of_twoPhaseChord", "OMAI.PhyslibBridge.Calphad.chemicalPotentials_equilibriumGibbs_of_twoPhaseChord", "OMAI.PhyslibBridge.Calphad.leverFractions_unique_and_bounds", "OMAI.PhyslibBridge.Calphad.transitionTemperature_unique" ], "source_path": "lean/bridge/OMAICalphad.lean", "build_record": "lean/bridge/build-evidence.json", "source_sha256": "6b8c4f275d905a5279e4203d3b0d872211344482103810c0134cc04e92f74b8e" } }, { "id": "canonical-rta-thermal-conductivity", "relation": "conditional", "statement": "With the relaxation-time mean free displacement F = v/(2 Gamma) and, as c, the actual Physlib canonical heat capacity of each harmonic mode at T > 0, the OMAI contraction kappa_ab = sum_m F_b c v_a/(N_q V_cell) is symmetric and positive semidefinite for positive linewidths and N_q V_cell > 0, and positive definite exactly when the group velocities of the modes span space.", "targets": [ { "kind": "node", "id": "ThermalConductivity[bte_solver=rta]", "uid": "d79d88d53266c147ea8d806cdbd30df309199469351ee13bcc9040a98745ebc4" }, { "kind": "operator", "id": "contract_kappa[bte_solver=rta]", "uid": "712b8145370a0072f225bd2f1d8c954b64311a3ca2f387320aa17e466926513c", "formula_sha256": "2e54bbf0cc370a56c0ffd43f2ca8cdefa185eb02d8d2388124e6586960adac5c" } ], "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.Basic", "name": "QuantumMechanics.OneDimension.HarmonicOscillator", "kind": "structure", "line": 63, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/Basic.lean#L63" } ], "conditions": [ "MeanFreeDisplacement[bte_solver=rta] is F_a = v_a/(2 Gamma) from solve_bte[bte_solver=rta], with Gamma the total half-width Im Sigma (phono3py's gamma), positive for every mode, so 1/tau = 2 Gamma. ShengBTE and kALDo report 1/tau, which must be halved before use as Gamma.", "HeatCapacity is, for each mode, the derivative-defined heat capacity of the full Nat harmonic-oscillator canonical ensemble of Physlib at the mode's positive angular frequency and T > 0, as in the harmonic-mode-heat-capacity binding. The oscillator mass does not enter.", "N_q and V_cell are positive; the mode sum is finite.", "Positive definiteness holds exactly when, for every nonzero direction x, some mode has v.x != 0. A crystal whose velocities all lie in a plane gives a semidefinite, singular kappa." ], "conventions": { "model_scope": "Relaxation-time contraction of independent harmonic modes with Physlib canonical heat capacities. Linewidths are positive inputs; no scattering model is verified.", "component_scope": "Full 3x3 Cartesian tensor kappa_ab; no isotropic reduction.", "normalization_basis": "Uniform mode sum divided by N_q V_cell with positive primitive-cell volume.", "frequency_convention": "Angular SI frequency in the canonical heat capacities. Gamma is the half-width Im Sigma in rad/s, with 1/tau = 2 Gamma. Stored linear THz requires omega_SI = 2*pi*10^12*nu_THz; velocities stored in Angstrom THz are already d omega/dq (1 Angstrom THz = 100 m/s).", "unit_embedding": "Coherent SI: c in J/K per mode, v in m/s, Gamma in rad/s, F in m, V_cell in m^3, kappa in W/(m K). These exact-real statements do not certify floating-point adapters.", "held_fixed": "Temperature, frequencies, velocities, linewidths, N_q and V_cell are fixed inputs.", "admissibility": "T > 0, every frequency positive, every linewidth positive, N_q V_cell > 0.", "reference_energy": "Zero-point energy does not enter the canonical heat capacity; no reference energy enters kappa.", "qpoint_weighting": "Uniform weights over a finite mode set; symmetry-reduced or tetrahedron weights are outside this definition." }, "remaining_obligations": [ "Linewidth producers (three-phonon, isotope, boundary) and their positivity are interpreted inputs, not verified.", "The full linearized-BTE kappa and the cumulative distributions are not covered; their catalog formulas are recorded as defects (see their gap reasons).", "Zero-frequency Gamma modes, which have no finite relaxation-time contribution, are excluded; convergence in the q mesh is not addressed." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibRTAConductivity", "declarations": [ "OMAI.PhyslibBridge.Lattice.rtaKappa_canonical" ], "source_path": "lean/bridge/OMAIPhyslibRTAConductivity.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_kappa[bte_solver=rta]" }, { "kind": "node", "id": "ThermalConductivity[bte_solver=rta]" } ], "source_sha256": "13267fa6d967df842f39251b23a0c2b70cd230f2553a444964f4bdc2229e34a8" } }, { "id": "cell-stress-rotation-balance", "relation": "local", "targets": [ { "kind": "node", "id": "Stress", "uid": "183e66d4e443326d2c346a0327cdb125026bd8f6a91a9bb97b61265fcade7e85" }, { "kind": "operator", "id": "compute_stress_cell", "uid": "88f8474bfda3d73e220f1db0ddfb8f00546007f1172ec663f4d4406568b3c5f2", "formula_sha256": "6dd196c0b221e9aed1dae7019b053498c5d5d96177025dc088250bca9555842f" } ], "statement": "For a cell energy E(F) of the homogeneously deformed cell, differentiable at F and invariant under rigid rotations of the deformed cell, the catalog stress sigma_ab = -(1/V) dE/d eps_ab satisfies sum_b sigma_{alpha b} F_{beta b} = sum_b sigma_{beta b} F_{alpha b} for every deformation gradient F; at the reference cell F = 1 the stress is symmetric, a corollary of the balance rather than a separate theorem.", "sources": [], "conditions": [ "The cell energy is a function of the deformation gradient F = 1 + eps, so derivatives with respect to the strain and to F agree, and V is the reference cell volume.", "The catalog stress is compressive positive, sigma = -(1/V) dE/d eps, the negative of the tension-positive Cauchy stress, as QE prints it in kbar.", "Rotation invariance of the deformed cell (objectivity) holds for any energy of the physical configuration; it is stated per rotation plane." ], "conventions": { "model_scope": "Exact derivative of one cell energy with respect to homogeneous strain; a local theorem over Mathlib with no Physlib model.", "component_scope": "Full 3x3 stress; the balance holds for each pair of Cartesian directions.", "normalization_basis": "Stress per unit reference volume.", "frequency_convention": "Not applicable; no frequencies enter these derivatives.", "unit_embedding": "Coherent SI: energy in J, volume in m^3, stress in Pa; 1 kbar = 1e8 Pa.", "held_fixed": "Internal atomic coordinates follow whatever the cell energy prescribes (clamped or relaxed); temperature is not involved.", "admissibility": "E differentiable at F; Invariance of the cell energy under rotations in the stated plane.", "reference_energy": "An additive constant in E changes no stress.", "qpoint_weighting": "Not applicable; no Brillouin-zone sums enter these derivatives." }, "remaining_obligations": [ "Pulay stress from a fixed plane-wave cutoff, k-point and smearing convergence, and the symmetrization codes apply to the computed tensor are not verified.", "The theorem concerns the exact strain derivative; finite-strain differences and Voigt-notation storage are not covered." ], "evidence": { "kind": "compiled_bridge", "module": "OMAICellElasticity", "declarations": [ "OMAI.PhyslibBridge.CellElasticity.cellStress_rotation_balance" ], "source_path": "lean/bridge/OMAICellElasticity.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_stress_cell" }, { "kind": "node", "id": "Stress" } ], "source_sha256": "c0663d570244e4c53317c1c7cbb193d5acd2429766316cc79685474bc7989e3d" } }, { "id": "classical-harmonic-frequency", "targets": [ { "kind": "node", "id": "Frequency", "uid": "fa262448f625b62f5a8c6934518bb8ddc5d3a5a5ce69203b3e2f6b10b21cc4b8" }, { "kind": "node", "id": "MolecularFrequency", "uid": "e1270ea9a293c9107226bf831f60985f5525a0c386dca70841743bdc5d082c26" } ], "relation": "conceptual", "statement": "The positive scalar spring oscillator frequency sqrt(k/m) is a restricted model related to harmonic vibrational frequencies; it is not a lattice dynamical-matrix or molecular Hessian identification.", "sources": [ { "module": "Physlib.ClassicalMechanics.HarmonicOscillator.Basic", "name": "ClassicalMechanics.HarmonicOscillator.ω", "kind": "def", "line": 156, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean#L156" }, { "module": "Physlib.ClassicalMechanics.HarmonicOscillator.Basic", "name": "ClassicalMechanics.HarmonicOscillator.ω_sq", "kind": "lemma", "line": 163, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean#L163" } ], "conditions": [ "Single oscillator spring constant k>0 and mass m>0." ], "conventions": { "model_scope": "One-dimensional positive-mass, positive-spring classical oscillator only.", "component_scope": "Scalar oscillator; no mode eigenvector or crystal index identification.", "normalization_basis": "not applicable", "frequency_convention": "Angular omega; linear THz and molecular wavenumber require explicit conversions.", "unit_embedding": "Consistent spring/mass/time units.", "held_fixed": "not applicable", "admissibility": "Stable positive frequency.", "reference_energy": "not applicable", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "Construct the mass-weighted mode or molecular-Hessian interpretation and prove the unit conversion before claiming a formula match." ], "evidence": { "kind": "source_correspondence" } }, { "id": "classical-harmonic-potential-force", "targets": [ { "kind": "node", "id": "Forces", "uid": "9c5a2fdedab6b668ccf5f2043fb4c29a982bcde5063fd6ac8633ee2c7858ecbf" }, { "kind": "node", "id": "Potential", "uid": "80a4273845e09207860e06db18d1f1ee5670a9b9c8b1020a99928c872878f901" } ], "relation": "conceptual", "statement": "Physlib harmonic potential and linear restoring force illustrate a restricted conservative-force model, not OMAI general interatomic potentials or a DFT Hellmann-Feynman theorem.", "sources": [ { "module": "Physlib.ClassicalMechanics.HarmonicOscillator.Basic", "name": "ClassicalMechanics.HarmonicOscillator.potentialEnergy", "kind": "def", "line": 200, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean#L200" }, { "module": "Physlib.ClassicalMechanics.HarmonicOscillator.Basic", "name": "ClassicalMechanics.HarmonicOscillator.force", "kind": "def", "line": 451, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean#L451" }, { "module": "Physlib.ClassicalMechanics.HarmonicOscillator.Basic", "name": "ClassicalMechanics.HarmonicOscillator.force_eq_linear", "kind": "lemma", "line": 464, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean#L464" } ], "conditions": [ "Single scalar spring k>0 and mass m>0; no claim for arbitrary atomistic potentials." ], "conventions": { "model_scope": "One-dimensional Hooke-law oscillator.", "component_scope": "One coordinate, not arbitrary atom-indexed force arrays.", "normalization_basis": "not applicable", "frequency_convention": "not applicable", "unit_embedding": "Consistent energy, length, and force units.", "held_fixed": "not applicable", "admissibility": "not applicable", "reference_energy": "Potential k*x^2/2 with reference at equilibrium.", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "Prove an atomistic restriction/coordinate interpretation before reusing these declarations as a model-specific OMAI result." ], "evidence": { "kind": "source_correspondence" } }, { "id": "composite-effective-medium", "relation": "local", "targets": [ { "kind": "node", "id": "DepolarizationFactor", "uid": "297b3254a09037f017769b771e19ad849443911048f77078a804cbc334c38309" }, { "kind": "node", "id": "ThermalConductivity[effective_medium=nan,orientation=aligned]", "uid": "fc292985123b709c76547dc02721e60797cf5693fbad40b874826291314d5dd3" }, { "kind": "node", "id": "ThermalConductivity[effective_medium=nan,orientation=random]", "uid": "8cee68745e003909cd8e65688628af53a35c0803ace3f330394bbf0edf338d7b" }, { "kind": "operator", "id": "depolarization_factors", "uid": "e8bd01e04d39ccad6f7046883bfa8e955e67c23fcc2f8868be045f327b18482f", "formula_sha256": "76cf8bc5f2cfa140cf9c69a817c7b4f0219d2b8336edf892f20184c6bfb304d4" }, { "kind": "operator", "id": "hasselman_johnson", "uid": "33e0be2a3db8fa86dce200159c6c0a273161ee6d1a49c75a369b1621197e0a1a", "formula_sha256": "9d0e588f6e47a7bc1e8345b77f6593376a9a9d3a07f847fd9ab69b039db49d6b" }, { "kind": "operator", "id": "nan_effective_kappa", "uid": "04d65e9f7a2f612eb94d018ebde08e8a1b39c1c5754d03e785ad92991a909fda", "formula_sha256": "11db4a2afb71060dd77bbbb7da720b27898064b0068e327550d1435f1f85783a" }, { "kind": "operator", "id": "nan_effective_kappa_aligned", "uid": "e86ac8a4b69dbb670dbe3f8ec7fda7ff0309670bac4713851ced64a061cd7962", "formula_sha256": "78db51dc39f38b3dff3f71c49db5450b2c9e89907cbbd61ba5b01852433353fb" } ], "statement": "Both closed-form branches of the catalog's spheroid depolarization factor and its sphere value equal int_0^1 t^2/(t^2 + p^2 (1 - t^2)) dt, so L33 lies in (0, 1), decreases strictly with the aspect ratio, is continuous through the sphere p = 1 with value 1/3, tends to 0 for needles and to 1 for disks. For every spheroid shape and volume fraction in [0, 1], the catalog's random Nan conductivity exceeds the matrix value by k_m f (2 b11 + b33)/(3 - f (2 L11 b11 + L33 b33)) with a positive denominator, so it exceeds k_m exactly when 2 b11 + b33 > 0, and filler whose Kapitza-corrected conductivities both exceed (both fall below) k_m raises (lowers) it. For spheres both Nan formulas reduce to Maxwell-Garnett with the Kapitza-corrected filler conductivity k/(1 + 2k/(G d)); without interface resistance (G growing without bound) the random sphere value tends to Maxwell-Garnett with the bulk filler conductivity, and the random value lies between the matrix and the corrected filler conductivity. The aligned axial Nan conductivity decreases strictly with L33 on [0, 1] and lies strictly between the series and parallel (Wiener) mixtures for L33 and f in (0, 1), so at fixed d3 it increases strictly with the aspect ratio. The Hasselman-Johnson formula is Maxwell-Garnett with the corrected conductivity at particle dimension d = 2 a_rad, so it agrees with the Nan sphere value under that radius-to-diameter conversion; it exceeds the matrix conductivity exactly above the critical radius (k_m/G) k/(k - k_m).", "sources": [], "conditions": [ "The filler is a dilute, monodisperse dispersion of identical axially symmetric spheroids in an isotropic matrix. Axis 3 is the symmetry axis and p = d3/d1 > 0, so p > 1 is prolate (fibre-like) and p < 1 is oblate (platelet-like). One interface conductance G_int > 0 applies to every face of every particle.", "G_int is the interface conductance q_int/Delta T_int with q_int the heat flux across the interface and Delta T_int > 0 the temperature drop in the direction of that flux, so G_int is positive; the formulas add the series resistance 1/G_int once at each of the two interfaces a particle presents along a direction.", "The catalog's d_1 and d_3 are full particle dimensions (the per-direction interface convention of Nan, Liu, Lin and Li, Appl. Phys. Lett. 85, 3549 (2004)), so a sphere has d equal to its diameter, while Hasselman-Johnson's a_rad is the sphere radius. The two formulas agree for spheres with d_1 = d_3 = 2 a_rad. Inserting the radius into d doubles the interface penalty 1/kc - 1/k = 2/(G_int d), from 1/(G_int a_rad) to 2/(G_int a_rad).", "k_f[1,1] is the filler conductivity transverse to axis 3 and k_f[3,3] the conductivity along axis 3 (in-plane and through-plane for a platelet). The sphere reductions require k_f[1,1] = k_f[3,3], d_1 = d_3 and L11 = L33 = 1/3; the Hasselman-Johnson formula reads only k_f[3,3].", "Random means isotropic three-dimensional orientation of the particles. Platelets oriented randomly within a plane are described by neither the random nor the aligned formula. The aligned operator is the component along the alignment axis 3, which is the through-plane direction only for platelets.", "f_vol is the filler volume fraction in [0, 1], not a mass fraction or a percentage. Conductivities, G_int and particle dimensions are positive and use one coherent unit system.", "Strict monotonicity of the aligned value in L33 and the strict Wiener bounds require 0 < f_vol < 1 and a Kapitza-corrected filler conductivity different from k_m; the aspect-ratio monotonicity varies p through L33 with d3, and so the axial Kapitza correction, held fixed. The critical-radius result requires a filler more conductive than the matrix and f_vol > 0." ], "conventions": { "model_scope": "Dilute effective-medium (Maxwell-Garnett, Nan, Hasselman-Johnson) conductivity of filled composites. Physlib has no effective-medium model, so these are local theorems about the OMAI formulas over Mathlib reals; no Physlib declaration is identified.", "component_scope": "Scalar random-orientation conductivity and the axial component of the aligned tensor; spheroidal inclusions with polar axis 3 only.", "normalization_basis": "Volume fraction of filler in the composite; conductivities are intensive.", "frequency_convention": "Not applicable; no frequencies enter these formulas.", "unit_embedding": "Coherent SI: conductivities in W/(m K), G_int in W/(m^2 K), lengths in m; f_vol, p and the depolarization factors are dimensionless.", "held_fixed": "Temperature, filler shape and orientation distribution, and the interface conductance are fixed; the theorems vary one of f_vol, L33, G_int or a_rad at a time as stated, and the aspect-ratio monotonicity varies p at fixed d3.", "admissibility": "p > 0; f_vol in [0, 1]; positive conductivities, lengths and G_int. Values outside these ranges have no asserted interpretation.", "reference_energy": "Not applicable.", "qpoint_weighting": "Not applicable." }, "remaining_obligations": [ "NEMD or measured producers of G_int and of the filler and matrix conductivities are not verified.", "Away from p = 1 the catalog's per-direction interface correction k/(1 + 2k/(G_int d)) (Nan et al. 2004) is a slab approximation with no validated accuracy; Nan et al., J. Appl. Phys. 81, 6692 (1997) uses K_p/(1 + R K_p L_ii (2 + 1/p)/a_1), and the two agree only at p = 1.", "The effective-medium formulas are dilute approximations. Particle interactions at high loading, percolation, clustering, porosity, polydisperse size or shape, and orientation distributions other than isotropic random or perfectly aligned are not covered.", "L11 = (1 - L33)/2 is the catalog's auxiliary sum rule and is not proved; the transverse aligned component is not proved.", "For non-spherical random filler no bound between the matrix and filler conductivities is proved beyond the sign result, and the Hashin-Shtrikman bounds are not proved.", "Anisotropic filler conductivity tensors beyond the two diagonal values, face-dependent interface conductance and temperature dependence are not covered." ], "evidence": { "kind": "compiled_bridge", "module": "OMAICompositeConductivity", "declarations": [ "OMAI.PhyslibBridge.Composites.depolarization_eq_integral", "OMAI.PhyslibBridge.Composites.depolarization_mem_Ioo", "OMAI.PhyslibBridge.Composites.depolarization_strictAntiOn", "OMAI.PhyslibBridge.Composites.depolarization_continuousOn", "OMAI.PhyslibBridge.Composites.tendsto_depolarization_atTop", "OMAI.PhyslibBridge.Composites.tendsto_depolarization_zero", "OMAI.PhyslibBridge.Composites.nanRandom_sub_matrix", "OMAI.PhyslibBridge.Composites.nanRandom_gt_iff", "OMAI.PhyslibBridge.Composites.nanRandom_sign", "OMAI.PhyslibBridge.Composites.nanRandom_sphere", "OMAI.PhyslibBridge.Composites.nanAligned_sphere", "OMAI.PhyslibBridge.Composites.nanRandom_sphere_between", "OMAI.PhyslibBridge.Composites.tendsto_nanRandom_sphere", "OMAI.PhyslibBridge.Composites.hasselmanJohnson_eq_maxwellGarnett", "OMAI.PhyslibBridge.Composites.hasselmanJohnson_critical_radius", "OMAI.PhyslibBridge.Composites.nanAligned_strictAntiOn", "OMAI.PhyslibBridge.Composites.nanAligned_strict_wiener", "OMAI.PhyslibBridge.Composites.nanAligned_strictMonoOn_aspect" ], "source_path": "lean/bridge/OMAICompositeConductivity.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "depolarization_factors" }, { "kind": "operator", "id": "hasselman_johnson" }, { "kind": "operator", "id": "nan_effective_kappa" }, { "kind": "operator", "id": "nan_effective_kappa_aligned" }, { "kind": "node", "id": "DepolarizationFactor" }, { "kind": "node", "id": "ThermalConductivity[effective_medium=nan,orientation=aligned]" }, { "kind": "node", "id": "ThermalConductivity[effective_medium=nan,orientation=random]" } ], "source_sha256": "e08799415b1e5345ed0cc6af0b192685f6d0f93606d3f8c0a0f91c3ed178096c" } }, { "id": "continuum-stress-sign", "targets": [ { "kind": "node", "id": "Pressure", "uid": "4ca6efdb874d146a6cca564c7a8a0504ee8868ae5890d4c33abddc147423d6f1" }, { "kind": "node", "id": "Stress", "uid": "183e66d4e443326d2c346a0327cdb125026bd8f6a91a9bb97b61265fcade7e85" }, { "kind": "operator", "id": "contract_pressure", "uid": "acbf580bd577b7499bc6806f2882511b90aaab32cabdb290a0de452d251e7514", "formula_sha256": "86f47f5fa90f900ae725b63de258ad5da3543e6ae3f07110e268241b7382edf3" } ], "relation": "conditional", "statement": "Under Physlib's three-dimensional inviscid stress law, OMAI's compressive-positive stress is minus the tensile-positive Cauchy stress, and trace(sigma_OMAI)/3 equals the given pressure at each spacetime point.", "sources": [ { "module": "Physlib.FluidDynamics.CauchyFlow.Inviscid", "name": "FluidDynamics.CauchyFlow.IsInviscid", "kind": "def", "line": 54, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/FluidDynamics/CauchyFlow/Inviscid.lean#L54" } ], "conditions": [ "Use dimension 3 and the upstream CauchyFlow.IsInviscid law stress_Physlib=-p*I.", "Set sigma_OMAI=-stress_Physlib in the same pressure units. Fin 3 axes 0,1,2 correspond to OMAI axes 1,2,3. No static, equilibrium or positive-pressure assumption is required." ], "conventions": { "model_scope": "Inviscid continuum stress, not an atomistic/DFT stress derivation.", "component_scope": "Three-dimensional stress tensor; explicit sign conversion.", "normalization_basis": "not applicable", "frequency_convention": "not applicable", "unit_embedding": "Pressure/stress in Pa after unit alignment.", "held_fixed": "One space-time operating point.", "admissibility": "Inviscid constitutive law at each spacetime point; no arbitrary deviatoric stress interpretation.", "reference_energy": "not applicable", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "The caller must justify the inviscid model and sign/unit interpretation. No DFT stress producer, mechanical equilibrium or thermodynamic free-energy derivative is proved." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibContinuum", "declarations": [ "OMAI.PhyslibBridge.compressiveStress_roundtrip", "OMAI.PhyslibBridge.compressiveStress_eq_pressure_identity", "OMAI.PhyslibBridge.compressiveStress_diagonal_of_isInviscid", "OMAI.PhyslibBridge.pressureFromStress_eq_physlib_pressure" ], "source_path": "lean/bridge/OMAIPhyslibContinuum.lean", "source_sha256": "e8bff3cb5d44cd78885d8c5f200d8761e8c568cd0d101fc0c2110f521bdca7ff", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_pressure" }, { "kind": "node", "id": "Pressure" } ] } }, { "id": "crystal-dos-landauer-conductance", "targets": [ { "kind": "node", "id": "PhononDOS", "uid": "e14b6cdd007ca3af971aa52880811138606efdb47e30d5a26c2335903ce796f3" }, { "kind": "node", "id": "ThermalConductance[transport_model=landauer]", "uid": "50084cef8405999bb14c2ff71b98f70f982d71254e3ec35bed543a10bbbbf542" }, { "kind": "operator", "id": "compute_conductance[transport_model=landauer]", "uid": "1201b09100e9aaac69f343b09ff8c8002c6de6dec1c843458bec89e3151ef436", "formula_sha256": "ae627a4fd4b58e0fea7656eb1b7f74fb8e4377ca53f2de6f080e09ca05e57aef" }, { "kind": "operator", "id": "compute_dos", "uid": "c42288d9daf6b41adc999cb7f35f8dc0d46478006f253df31c7aedf2605783bc", "formula_sha256": "8c3bd346a566aedaa4ea5dc0a812a8404cd1a6e7d221e4e7e5709550addeab01" } ], "relation": "conditional", "statement": "compute_dos is read as the measure g = (1/N_q) sum_{q,nu} delta(omega - omega_{q nu}) on frequency. On a finite Born-von Karman crystal of Physlib sites, with the frequencies the nonnegative square roots of the eigenvalues of the catalog dynamical matrix built from the crystal's own force constants, taken at every character of the finite translation group (the full mesh, N_q = |G|), a harmonic energy that is never negative and positive masses give the sum rules int g = 3 N_cell, the number of modes per wavevector, and int omega^2 g = sum_i Phi_ii(0)/M_i over the unit-cell displacement components, so the mean-square frequency is the mass-weighted on-site force constant averaged over modes. compute_conductance[transport_model=landauer] is read as G(T) = (1/(2 pi)) int_0^infinity hbar omega T(omega) d n_BE(hbar omega/(k_B T))/dT domega. For omega > 0 and T > 0 the weight hbar omega dn_BE/dT is the derivative-defined canonical heat capacity of the Physlib quantum harmonic oscillator of angular frequency omega, with Physlib's hbar and k_B, so G(T) is the transmission-weighted spectral integral of the Physlib heat capacity of one oscillator per frequency divided by 2 pi; with unit transmission G(T) = pi^2 k_B^2 T/(3 h), the thermal conductance quantum, from int_0^infinity x^2 e^x/(e^x - 1)^2 dx = pi^2/3.", "sources": [ { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal", "kind": "structure", "line": 65, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L65" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.Basic", "name": "QuantumMechanics.OneDimension.HarmonicOscillator", "kind": "structure", "line": 63, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/Basic.lean#L63" }, { "module": "Physlib.Thermodynamics.Temperature.Basic", "name": "Temperature", "kind": "structure", "line": 32, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/Thermodynamics/Temperature/Basic.lean#L32" } ], "conditions": [ "compute_dos is transcribed as a sum of point masses (1/N_q) sum_{q,nu} delta(omega - omega_{q nu}); the catalog's Gaussian or tetrahedron dos_broadening and symmetry-reduced meshes are not covered. The crystal is Physlib.CondensedMatter.Crystal with a finite translation group G, as in the crystal-phonon-dispersion binding: the mesh is every character of G (N_q = |G|, the full Born-von Karman mesh) and the frequencies are the nonnegative square roots of the eigenvalues of the catalog dynamical matrix built from the supercell's own force constants with the cell-vector phase.", "The total-weight rule int g = 3 N_cell needs only the mode count (three per unit-cell site) and holds for any frequencies; the second-moment rule needs a harmonic energy that is never negative (Stable, so every eigenvalue is nonnegative and omega^2 is the eigenvalue) and positive masses, and reads sum_i Phi_ii(0)/M_i over the 3 N_cell displacement components i of the reference cell, Phi_ii(0) the on-site self force constant. Zero modes count at omega = 0.", "compute_conductance[transport_model=landauer] is transcribed with the transmission T(omega) at angular frequency omega and the derivative in T of n_BE(hbar omega/(k_B T)). The weight identity hbar omega dn_BE/dT = c_omega(T) holds for omega > 0 and T > 0, with c_omega the derivative-defined heat capacity of Physlib's canonical ensemble of the full Nat-spectrum oscillator of angular frequency omega, as in the harmonic-mode-heat-capacity binding, and with Physlib's hbar and k_B.", "The integral form G = (1/(2 pi)) int T(omega) c_omega(T) domega is an integrand identity under the Lebesgue integral over (0, infinity) and does not by itself establish integrability for an arbitrary T(omega); the unit-transmission value pi^2 k_B^2 T/(3 h), 9.46e-13 W/K at 1 K, is proved with the explicit Bose integral. Ballistic, coherent transport in the Landauer picture: the transmission is an input and its producers are not verified.", "Frequencies are angular (rad/s); T(omega) is a dimensionless number of transmitting channels at omega; hbar in J s and k_B in J/K give G in W/K. Physlib's h is 2 pi hbar." ], "conventions": { "model_scope": "Harmonic phonon density of states on a finite Physlib crystal, and the Landauer thermal conductance with the Physlib canonical heat capacity of one quantum oscillator per frequency. The lattice-dynamics and Landauer content is OMAI-local; Physlib supplies the crystal bookkeeping, the oscillator, the canonical heat capacity and the constants.", "component_scope": "Scalar density of states on frequency and scalar conductance of one lead pair; no tensor components.", "normalization_basis": "Density of states per wavevector (divided by N_q) with 3 N_cell modes per wavevector; conductance per channel through T(omega).", "frequency_convention": "Angular SI frequency omega in rad/s for both formulas, as in the canonical heat capacities; stored linear THz requires omega = 2 pi 10^12 nu, and the density of states then rescales by 2 pi 10^12.", "unit_embedding": "Coherent SI: omega in rad/s, Phi in N/m, M in kg, hbar in J s, k_B in J/K, T in K, G in W/K. Exact-real statements; floating-point evaluation is not certified.", "held_fixed": "Force constants, masses and the translation group for the density of states; the transmission and the temperature for the conductance.", "admissibility": "Finite translation group and positive masses; Stable force constants for the second moment; omega > 0 and T > 0 for the weight identity; T > 0 for the conductance quantum.", "reference_energy": "Zero-point energy does not enter the canonical heat capacity or the conductance.", "qpoint_weighting": "Uniform weights over every character of the finite translation group; symmetry-reduced or tetrahedron weights are outside these statements." }, "remaining_obligations": [ "Smeared and tetrahedron densities of states, symmetry-reduced meshes, interpolation to incommensurate wavevectors and the phonopy, kALDo and Quantum ESPRESSO producers are not verified; higher moments and the Debye model are not covered.", "The transmission function (atomistic Green's function or mode counting) and its producers are not verified; integrability of the conductance integrand for an arbitrary transmission is not proved; inelastic and diffusive corrections to the Landauer picture are outside the model.", "The Physlib constants hbar and k_B are used as declared; no statement is made about their values beyond Physlib's definitions." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibPhononSpectrum", "declarations": [ "OMAI.PhyslibBridge.PhononSpectrum.crystalDOS_sum_rules", "OMAI.PhyslibBridge.PhononSpectrum.landauerWeight_eq_heatCapacity", "OMAI.PhyslibBridge.PhononSpectrum.catalogLandauer_eq_heatCapacity", "OMAI.PhyslibBridge.PhononSpectrum.catalogLandauer_unit_transmission" ], "source_path": "lean/bridge/OMAIPhyslibPhononSpectrum.lean", "build_record": "lean/bridge/build-evidence.json", "source_sha256": "ccc6127cddcc1efcf6c24516648dfff1dd801d9b099000c9a0f1ce55305fde3b" } }, { "id": "crystal-dynamical-matrix", "relation": "conditional", "statement": "For second-order force constants on the sites of a Physlib crystal with equal mixed partials, translation invariance and finite range, the index-permutation symmetry Phi_ij(-R) = Phi_ji(R) follows from the translation invariance, the OMAI dynamical matrix D_ij(q) = sum_R e^{i q.R} Phi_ij(R)/sqrt(M_i M_j) is Hermitian at every wavevector, and with the acoustic sum rule the Gamma-point matrix has at least three independent zero modes, the mass-weighted rigid translations.", "targets": [ { "kind": "node", "id": "BareDynamicalMatrix", "uid": "e62e58716111fdc1dff24d18e75c70904cbb07ce6e0025e24fcdac8c4ed0a53d" }, { "kind": "operator", "id": "compute_dynamical_matrix", "uid": "67d4d9c1ad01e2aa1705e54ed9edb2d1a0ce601aa0e7c9929a7ba7cbd7259f28", "formula_sha256": "d084d4d043a870f7a1d7dcb9931a150768819a4ce1f993c3620ad5b8e99e8897" } ], "sources": [ { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal", "kind": "structure", "line": 65, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L65" }, { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal.Site", "kind": "structure", "line": 90, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L90" } ], "conditions": [ "Represent the crystal with Physlib.CondensedMatter.Crystal: a finitely generated abelian translation group, a finite nonempty unit cell and sites (cell, label), on which the translation action moves the cell and keeps the label. A mode index is a unit-cell label with a Cartesian axis, and Phi_ij(R) is the reference-cell component i against component j in cell R.", "Force constants are real second derivatives of the lattice potential with equal mixed partials, invariant under every lattice translation, and the reference-cell interactions vanish outside a finite set of cells, so the catalog sum over all R is finite.", "The phase e^{i R q} is exp(i q.pos(R)) for an additive embedding pos of the translations into Cartesian space, the catalog's cell-vector convention. Masses are positive and per unit-cell label.", "The acoustic sum rule, sum over cells and labels of Phi_{(kappa,alpha),(kappa',beta)}(R) = 0, is a stated hypothesis for the Gamma-point result. Phonopy and other producers enforce it numerically; it is not derived here from a potential." ], "conventions": { "model_scope": "Harmonic lattice dynamics. Physlib's Crystal supplies the site and translation bookkeeping (sites, cells, the translation action and the site equivalence); the force-constant and dynamical-matrix content is OMAI-local. No anharmonic, non-analytic, electronic or numerical producer is modeled.", "component_scope": "Mode index (unit-cell label, Cartesian axis); one dynamical matrix per wavevector, with all tensor components kept.", "normalization_basis": "Mass weighting 1/sqrt(M_i M_j) with masses per unit-cell label; the lattice sum runs over the finite set of interacting cells closed under inversion.", "frequency_convention": "Eigenvalues are angular omega^2 in (rad/s)^2 with force constants in J/m^2 and masses in kg. Stored linear THz frequencies require omega_SI = 2*pi*10^12*nu_THz. Group velocities stored in Angstrom THz are already d omega/dq, 1 Angstrom THz = 100 m/s, with no factor 2*pi.", "unit_embedding": "Coherent SI: force constants in J/m^2 (1 eV/Angstrom^2 = 16.02176634 J/m^2), masses in kg, lattice vectors in m and wavevectors in rad/m. These exact-real statements do not certify floating-point adapters.", "held_fixed": "Crystal, unit cell, masses, force constants and the lattice embedding are fixed; only the wavevector varies.", "admissibility": "Positive masses; real force constants with equal mixed partials, translation invariance and finite range. Phase convention: the catalog and the Lean definitions use the cell-vector phase e^{+i q.R}. Phonopy and phono3py use atom positions, D_ph = U^dagger D_cat U with U = diag(e^{i q.tau_kappa}). omega^2 and nondegenerate velocities agree; eigenvectors differ by e^{-i q.tau} and off-diagonal velocity-operator elements differ, so phonopy eigenvectors fed into the catalog's dD/dq give wrong velocities.", "reference_energy": "The static lattice energy and any constant potential offset do not enter second derivatives.", "qpoint_weighting": "No q-point weighting is involved; the Hermiticity holds at every wavevector and the zero modes at Gamma." }, "remaining_obligations": [ "Finite-displacement, DFPT and potential-based force-constant producers (compute_force_constants[order=2], compute_fc2_finite_displacement) are interpreted inputs, not verified implementations; supercell folding of computed force constants and their symmetrization are not modeled.", "The non-analytic correction apply_nac_correction is covered by the static-dielectric binding in its q -> 0 limit only; the finite-q Ewald sums are not.", "Long-range force constants without a finite range, Ewald-summed dipole tails and the thermodynamic limit of an infinite sum are outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibLatticeDynamics", "declarations": [ "OMAI.PhyslibBridge.Lattice.ForceConstants.fc_neg", "OMAI.PhyslibBridge.Lattice.ForceConstants.dynamicalMatrix_isHermitian", "OMAI.PhyslibBridge.Lattice.ForceConstants.three_le_finrank_ker_gamma" ], "source_path": "lean/bridge/OMAIPhyslibLatticeDynamics.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_dynamical_matrix" }, { "kind": "node", "id": "BareDynamicalMatrix" } ], "source_sha256": "79095baa161d4b068e4a7cb1a5a00123dc4743b03463b6157a2bf3c906839f36" } }, { "id": "crystal-group-velocity", "relation": "conditional", "statement": "On a differentiable eigenvector path, away from Gamma: for the dynamical matrix of a Physlib crystal and an eigenvector path along the wavevector line q + s e_alpha that is differentiable at s = 0, normalized with real eigenvalue omega^2(s) for s near 0, and has omega^2 > 0 at s = 0, the eigenvalue is differentiable and the OMAI group velocity e^dagger (dD/dq_alpha) e / (2 omega), with dD/dq_alpha the actual derivative of the catalog dynamical matrix, is a real number equal to d omega/d q_alpha (Hellmann-Feynman). A compiled monatomic chain gives the textbook a sqrt(K/M) cos(q a / 2).", "targets": [ { "kind": "node", "id": "GroupVelocity", "uid": "d1934e166dbd0c91125936f9b2ccc0100af0ba44ed439003aae76b47b6d26fc6" }, { "kind": "operator", "id": "compute_group_velocity", "uid": "3cf7b8aa48dca910c062ce17738ede9cb609bacd7bd685bae23ac698897e38c3", "formula_sha256": "0b3f5977eb1cc8e4d87d72b54acc6ad49ac808800d4c3fbe88b7e9408ab41887" } ], "sources": [ { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal", "kind": "structure", "line": 65, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L65" }, { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal.Site", "kind": "structure", "line": 90, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L90" } ], "conditions": [ "DynamicalMatrix is the crystal dynamical matrix of the crystal-dynamical-matrix binding without the non-analytic correction, and dD/dq_alpha is its exact derivative sum_R i R_alpha e^{i q.R} Phi_ij(R)/sqrt(M_i M_j) along the Cartesian axis alpha.", "Along the line q + s e_alpha the eigenvector e(s) is differentiable at s = 0 and, on a neighbourhood of s = 0, normalized with real eigenvalue omega^2(s); omega^2(0) > 0. Nothing is assumed about e(s) outside that neighbourhood. A nondegenerate branch away from Gamma admits such a path by analytic perturbation theory; that existence is a hypothesis. Symmetry-enforced degeneracies are generic (transverse pairs along Gamma-X and Gamma-L), and at such points the eigenvector a code returns is not on the path: phonopy projects the velocity per direction and phono3py averages over degenerate sets and the q-star.", "omega = sqrt(omega^2) is angular frequency; the velocity is in m/s when q is in rad/m. Velocities stored in Angstrom THz are already d omega/dq (1 Angstrom THz = 100 m/s, no 2*pi). The eigenvectors must be in the catalog's cell-vector phase convention.", "The compiled monatomic chain (Physlib ofUnitCell Z Unit, nearest-neighbour longitudinal springs K, mass M, spacing a) has omega^2 = (2K/M)(1 - cos(q_x a)) and catalog velocity a sqrt(K/M) cos(q_x a / 2) on 0 < q_x a < pi. It is an exact model example, not a material fit." ], "conventions": { "model_scope": "Harmonic lattice dynamics. Physlib's Crystal supplies the site and translation bookkeeping (sites, cells, the translation action and the site equivalence); the force-constant and dynamical-matrix content is OMAI-local. No anharmonic, non-analytic, electronic or numerical producer is modeled.", "component_scope": "Mode index (unit-cell label, Cartesian axis); one dynamical matrix per wavevector, with all tensor components kept.", "normalization_basis": "Mass weighting 1/sqrt(M_i M_j) with masses per unit-cell label; the lattice sum runs over the finite set of interacting cells closed under inversion.", "frequency_convention": "Eigenvalues are angular omega^2 in (rad/s)^2 with force constants in J/m^2 and masses in kg. Stored linear THz frequencies require omega_SI = 2*pi*10^12*nu_THz. Group velocities stored in Angstrom THz are already d omega/dq, 1 Angstrom THz = 100 m/s, with no factor 2*pi.", "unit_embedding": "Coherent SI: force constants in J/m^2 (1 eV/Angstrom^2 = 16.02176634 J/m^2), masses in kg, lattice vectors in m and wavevectors in rad/m. These exact-real statements do not certify floating-point adapters.", "held_fixed": "Crystal, unit cell, masses, force constants and the lattice embedding are fixed; only the wavevector varies.", "admissibility": "Positive masses; real force constants with equal mixed partials, translation invariance and finite range. Phase convention: the catalog and the Lean definitions use the cell-vector phase e^{+i q.R}. Phonopy and phono3py use atom positions, D_ph = U^dagger D_cat U with U = diag(e^{i q.tau_kappa}). omega^2 and nondegenerate velocities agree; eigenvectors differ by e^{-i q.tau} and off-diagonal velocity-operator elements differ, so phonopy eigenvectors fed into the catalog's dD/dq give wrong velocities.", "reference_energy": "The static lattice energy and any constant potential offset do not enter second derivatives.", "qpoint_weighting": "No q-point weighting is involved; the result holds along each differentiable eigenvector path." }, "remaining_obligations": [ "Finite-difference velocity estimators, degenerate-subspace treatments and symmetry-averaged velocities in phonopy, phono3py, kaldo and ShengBTE are not verified.", "The existence of a differentiable eigenvector path on a nondegenerate branch is not proved here.", "NAC-corrected velocities near Gamma and velocities at band crossings are outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibGroupVelocity", "declarations": [ "OMAI.PhyslibBridge.Lattice.hellmann_feynman", "OMAI.PhyslibBridge.Lattice.ForceConstants.catalogGroupVelocity_eq_deriv" ], "source_path": "lean/bridge/OMAIPhyslibGroupVelocity.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_group_velocity" }, { "kind": "node", "id": "GroupVelocity" } ], "source_sha256": "d9d14bdfc6c429bd5ee07f85e659bdb2a56fc208b7fc6356cbf4ac76df495dca" } }, { "id": "crystal-phonon-dispersion", "relation": "conditional", "statement": "Let N be the sublattice of a finite Born-von Karman supercell of a Physlib crystal and fold the crystal force constants into the supercell, Phi_BvK(R) = sum_L Phi(R + L). At every wavevector q commensurate with the supercell (e^{i q.L} = 1 for every L in N), the supercell dynamical matrix at the corresponding character equals the OMAI dynamical matrix D(q). If the folded supercell is mechanically stable (its harmonic energy is never negative), D(q) is positive semidefinite there, so every solution of the OMAI dispersion relation with a nonzero eigenvector has real omega^2 >= 0 and a real frequency.", "targets": [ { "kind": "operator", "id": "compute_dispersion", "uid": "1ab7a3aee9b25b786739afcd517b909e4fdf091c768ab4ecbea007e53633be92", "formula_sha256": "58532a304fabac6818bbb184abfe0a1f3b140583aec52ace8a1b4a73f1d8db5e" } ], "sources": [ { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal", "kind": "structure", "line": 65, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L65" }, { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal.Site", "kind": "structure", "line": 90, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L90" }, { "module": "Physlib.CondensedMatter.Crystal.Basic", "name": "Physlib.CondensedMatter.Crystal.siteEquiv", "kind": "def", "line": 97, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Crystal/Basic.lean#L97" } ], "conditions": [ "DynamicalMatrix is the crystal dynamical matrix of the crystal-dynamical-matrix binding, passed through identity_dm without the non-analytic correction, under the same force-constant conditions.", "The supercell is the quotient of the translations by a sublattice N with finitely many cosets (periodic boundary conditions). Its force constants are the folded sums of the crystal force constants; mechanical stability is the hypothesis that the real harmonic energy sum u_a Phi_BvK,ab u_b of the supercell is never negative.", "Only wavevectors commensurate with the supercell are covered. Incommensurate meshes are the normal case for BTE (for example a 4x4x4 force-constant supercell with a 19x19x19 kappa mesh), and Fourier interpolation of supercell force constants is exact only at commensurate q; nothing is stated at the other mesh points.", "omega is the nonnegative square root of omega^2 on a stable supercell; imaginary modes of an unstable structure are excluded by the hypothesis, not removed. Two compiled witnesses: Einstein springs on ofUnitCell (ZMod 3) Unit satisfy the supercell stability hypothesis directly, and the monatomic chain's nearest-neighbour springs folded onto the three-cell ring Z/3Z (springs_fold_three_stable, with springs_fold_stable for every finite ring Z/N) satisfy the folded-stability hypothesis of the commensurate-wavevector theorem." ], "conventions": { "model_scope": "Harmonic lattice dynamics. Physlib's Crystal supplies the site and translation bookkeeping (sites, cells, the translation action and the site equivalence); the force-constant and dynamical-matrix content is OMAI-local. No anharmonic, non-analytic, electronic or numerical producer is modeled.", "component_scope": "Mode index (unit-cell label, Cartesian axis); one dynamical matrix per wavevector, with all tensor components kept.", "normalization_basis": "Mass weighting 1/sqrt(M_i M_j) with masses per unit-cell label; the lattice sum runs over the finite set of interacting cells closed under inversion.", "frequency_convention": "Eigenvalues are angular omega^2 in (rad/s)^2 with force constants in J/m^2 and masses in kg. Stored linear THz frequencies require omega_SI = 2*pi*10^12*nu_THz. Group velocities stored in Angstrom THz are already d omega/dq, 1 Angstrom THz = 100 m/s, with no factor 2*pi.", "unit_embedding": "Coherent SI: force constants in J/m^2 (1 eV/Angstrom^2 = 16.02176634 J/m^2), masses in kg, lattice vectors in m and wavevectors in rad/m. These exact-real statements do not certify floating-point adapters.", "held_fixed": "Crystal, unit cell, masses, force constants and the lattice embedding are fixed; only the wavevector varies.", "admissibility": "Positive masses; real force constants with equal mixed partials, translation invariance and finite range. Phase convention: the catalog and the Lean definitions use the cell-vector phase e^{+i q.R}. Phonopy and phono3py use atom positions, D_ph = U^dagger D_cat U with U = diag(e^{i q.tau_kappa}). omega^2 and nondegenerate velocities agree; eigenvectors differ by e^{-i q.tau} and off-diagonal velocity-operator elements differ, so phonopy eigenvectors fed into the catalog's dD/dq give wrong velocities.", "reference_energy": "The static lattice energy and any constant potential offset do not enter second derivatives.", "qpoint_weighting": "Commensurate wavevectors of the supercell only: the characters of the translations modulo N. No mesh weights enter." }, "remaining_obligations": [ "Numerical eigensolvers, branch sorting and band-crossing tracking in phonopy, kaldo or ShengBTE are not verified.", "Mechanical stability of a particular material and the folding of computed supercell force constants are hypotheses; the theorem does not certify that a computed structure is a minimum.", "Incommensurate wavevectors, Fourier interpolation between commensurate points, the infinite-crystal limit of stability and NAC-corrected matrices are outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibPhononDispersion", "declarations": [ "OMAI.PhyslibBridge.Lattice.ForceConstants.dynamicalMatrix_posSemidef_of_stable", "OMAI.PhyslibBridge.Lattice.ForceConstants.fold_dynamicalMatrix", "OMAI.PhyslibBridge.Lattice.ForceConstants.planeWave_dynamicalMatrix_posSemidef_of_folded_stable", "OMAI.PhyslibBridge.Lattice.dispersion_omega2_nonneg" ], "source_path": "lean/bridge/OMAIPhyslibPhononDispersion.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_dispersion" } ], "source_sha256": "70bd052d1d32afe828a3a8f7e26e406d5ec1e121b86ad0d9b08717f8c0f87f1a" } }, { "id": "dft-stability-energy-differences", "relation": "local", "targets": [ { "kind": "node", "id": "AdsorptionEnergy", "uid": "949582fba59372799c672b38510bd07f2e733f8bd6e63887b9728a672b1c5b25" }, { "kind": "node", "id": "EnergyAboveHull", "uid": "910d47d67a7308728e2f7d211a08c1590239c7b802cd75128f00cd103a719b84" }, { "kind": "node", "id": "GrainBoundaryEnergy", "uid": "f81c9c7907dc4297e44624098da25bfd87c1654a4198853bb185891282c24528" }, { "kind": "node", "id": "ReactionEnergy", "uid": "a5736aa4760c27562f9e189829763a891b6fed83fe4a030a7e306b8d909d825c" }, { "kind": "node", "id": "SurfaceEnergy", "uid": "306384aed4a1204fb7ca2347729de8d1816fa5fc32f4855a5d5f56a4d0fc1ddf" }, { "kind": "node", "id": "Voltage", "uid": "26831a0aabc3508a944405ec2fd763a0637abeeac1ed3d87df64c27392d638f4" }, { "kind": "operator", "id": "compute_adsorption_energy", "uid": "35f53c511ef6202b79b33e89856071aa1f3f8610ad7705bd3f608784d6216a1f", "formula_sha256": "55f4d4ba2cc61605a32dbe38ea3b7883519a192efc4d173bca1a654bd09810f0" }, { "kind": "operator", "id": "compute_energy_above_hull", "uid": "4e7f15bf40f88fab20271ac052a22b491e7955ab3ebab9bb229713adf057a040", "formula_sha256": "8c6556dbca3a51a285840593e5ab35fa13c898956188a1beec9febac45ae722b" }, { "kind": "operator", "id": "compute_grain_boundary_energy", "uid": "74ba9279dc65f3acd9e2be4ca13861b510212aa24a19aca4b29c65976003bf1b", "formula_sha256": "f9a65c5a348516a72f064663e00f68d094e9511ed620539de392d4377397aca6" }, { "kind": "operator", "id": "compute_intercalation_voltage", "uid": "a4bc99a6b2efc1df1400041ada33cfb59e97819f189e3501a99c06f9c5e5d49f", "formula_sha256": "7a3dd261cb0bf1bb623ff3a0c3a2a353a03573911235a56c05e01e021b6f4949" }, { "kind": "operator", "id": "compute_reaction_energy", "uid": "2204ada808d0fae7706fe31723755bcc9614f2901bff9230e163f47a7e95cdf7", "formula_sha256": "c69c9c84d6d962cea6ea1f005df0612832d065d6fa4d759dfff5c9b1e08cbe1f" }, { "kind": "operator", "id": "compute_surface_energy", "uid": "c8b2e006408e3e2c08f6c71eb7cd4e1a07f72ab20034fe883c691c1cad278492", "formula_sha256": "c99d3c4dab85511ea00104a21376fad96e0f80ce1982fddad3ac9f531454e8eb" } ], "statement": "The interpreted hull distance of per-atom formation energies is the same for every set of elemental reference energies, and more generally does not change when every competing entry's formation energy is shifted by the same affine function of composition; a supporting affine function of composition that touches an entry certifies a zero distance. The adsorption energy and the average intercalation voltage are unchanged by every per-element energy offset exactly when the cells conserve atoms. The average intercalation voltage is the mean of the open-circuit voltage profile -(E'(x) - mu)/q over the ion content, also for a convex-hull cell energy with kinks between voltage plateaus. For slab energies affine in the number of bulk units, the catalog's surface energy, and its grain-boundary energy, which is the same function (grainBoundaryEnergy_eq_surfaceEnergy), are independent of slab thickness exactly when the bulk reference equals the slab slope, equal intercept/(2A) then, and grow without bound with thickness otherwise. The catalog's reaction energy sum_i c_i N_i Delta H_f,i, the signed formula-unit coefficients times the atoms of each formula unit times the per-atom formation energies, is independent of the elemental reference energies exactly when the reaction is balanced.", "sources": [], "conditions": [ "compute_energy_above_hull is opaque in the catalog and is interpreted as the per-atom distance of the entry's formation energy above the lower convex hull of the competing entries, the hull being the least weighted formation energy of nonnegative weights on the competing set that sum to one and reproduce the composition in atomic fractions. The entry belongs to the competing set. The distance is zero for every entry on the lower convex hull, not only for a unique ground state.", "The target and its competitors share one energy model and one correction scheme; an MLIP candidate against MP2020-corrected DFT competitors is not covered. The affine invariance covers a change of the elemental reference energies and any per-atom correction c + sum_a delta_a x_a applied to every entry alike. The MP2020 scheme is not such a shift (its correction class depends on the U value and the anion type, and elemental entries are uncorrected), so the theorem applies only within one correction class.", "Adding competitors can only raise the hull distance, so a value against an incomplete competing set is a lower bound of the value against a complete one, provided the entry energies do not depend on the competing set; MaterialsProjectDFTMixingScheme recomputes R2SCAN energies from the whole entry set and breaks this reading.", "The adsorption energy uses three cells from one energy model and one calculator setting. The three-term catalog formula cannot express fractional gas-phase references (1/2 E_O2, E_H2O - E_H2), has no per-adsorbate normalization for a coverage theta, and needs a dipole correction for one-sided slabs.", "The intercalation voltage sign follows the catalog: V = -(E_full - E_empty - n mu)/(n q), positive when inserting n ions from the metal lowers the energy, with q the charge transferred per ion. The catalog's n_ion q_e hard-codes one electron per ion, which is correct for Li, Na and K; for an ion of valence z the transferred charge is n z e, so q = z e; the catalog restricts its form to monovalent working ions and states the n z e rule for valence z. The theorems hold for any q != 0. The mean-voltage theorem needs E(x) continuous on [0, n] and differentiable off a countable set with an integrable derivative, which admits the kinks of a convex-hull energy with several plateaus.", "The surface-energy formula needs a stoichiometric slab with two equivalent, equally relaxed terminations; otherwise gamma depends on the chemical potentials of the excess species (pymatgen SlabEntry). The grain-boundary form with 2A holds only for a fully periodic bicrystal with two equivalent boundaries; with vacuum the cell holds one boundary and two free surfaces, and relax_cell=False biases gamma_GB high. Both thickness results assume slab energies affine in the number N of bulk units, which holds once the two surfaces or boundaries no longer interact.", "compute_formation_energy is a per-atom definition with nothing to prove on its own and is not a proof subject; its reference set drops out of the hull distance. compute_reaction_energy, corrected on 2026-10-29 to sum_i c_rxn[i] N^fu[i] dH_f[i], is transcribed as catalogReactionEnergy with N^fu[i] the atoms of the formula unit whose per-atom formation energy is dH_f[i]; it is independent of the elemental references exactly when the reaction is balanced, sum_i c_rxn[i] n_{i,a} = 0 for every element a (catalogReactionEnergy_reference_independent_iff), and the balance condition is a hypothesis, not checked by the formula." ], "conventions": { "model_scope": "Ground-state energy differences built from cell energies of one electronic-structure or interatomic model. Physlib has no model of convex hulls, slabs or electrodes, so these are local theorems over Mathlib reals that identify no Physlib declaration.", "component_scope": "Finite competing sets and finite element sets; cells are described by their atom counts per element. Slab thickness is the number of bulk units; the in-plane area is fixed.", "normalization_basis": "Formation energies and hull distances per atom; adsorption energies per adsorbate configuration; voltages per transferred charge q per ion; surface and grain-boundary energies per unit area of one surface or boundary.", "frequency_convention": "Not applicable; no frequencies enter these formulas.", "unit_embedding": "Energies in eV or J in one embedding throughout, areas in A^2 or m^2 (1 eV/A^2 = 16.021766 J/m^2), charge in elementary charges so that eV/e is one volt.", "held_fixed": "The energy model, its settings and the elemental reference energies are common to every cell entering one difference; the competing set is fixed for one hull; the slab area is fixed across thicknesses.", "admissibility": "The entry belongs to its competing set; per-atom formation energies need a nonzero atom count; the intercalation iff needs n != 0 and q != 0; the thickness criterion needs a nonzero area.", "reference_energy": "Elemental reference chemical potentials enter formation energies and drop out of hull distances and of balanced reaction energies. The working-ion reservoir mu is the energy per atom of the metal in the same model.", "qpoint_weighting": "Not applicable; no Brillouin-zone sums enter these formulas." }, "remaining_obligations": [ "The DFT or interatomic total energies, relaxations and structure selection of compute_formation_energy's reference set, the hull_source competing set, slab and grain-boundary construction and the adsorption site are not verified, nor the MP2020 or GGA/GGA+U mixing corrections, which are not common affine shifts of every entry.", "Only the sufficiency of a supporting affine function for a zero hull distance is proved; its necessity, which needs convex duality, is not.", "Finite-size slab effects beyond the affine regime (surface-surface interaction in thin slabs, dipole corrections for asymmetric or polar slabs) and non-stoichiometric slabs are not covered.", "Finite-temperature, vibrational, configurational and electrolyte contributions to stability, voltage and adsorption are not covered; the energies are ground-state values.", "The catalog's intercalation-voltage form is restricted to monovalent working ions; the theorems here take q as stated in the conditions, so a multivalent ion needs q = z e supplied. Whether a reaction is balanced, and the temperature of the rxn-network dGf(T) entries, are not checked." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIStabilityEnergies", "declarations": [ "OMAI.PhyslibBridge.Stability.energyAboveHull_formationEnergy_reference_independent", "OMAI.PhyslibBridge.Stability.energyAboveHull_affine_invariant", "OMAI.PhyslibBridge.Stability.energyAboveHull_eq_zero_of_supporting", "OMAI.PhyslibBridge.Stability.adsorptionEnergy_offset_invariant_iff", "OMAI.PhyslibBridge.Stability.intercalationVoltage_offset_invariant_iff", "OMAI.PhyslibBridge.Stability.intercalationVoltage_eq_mean_ocv", "OMAI.PhyslibBridge.Stability.surfaceEnergy_thickness_independent_iff", "OMAI.PhyslibBridge.Stability.catalogReactionEnergy_reference_independent_iff" ], "source_path": "lean/bridge/OMAIStabilityEnergies.lean", "build_record": "lean/bridge/build-evidence.json", "source_sha256": "b0fd879de1d46103e675413995828b25254511988335990bb901354b6dd2ca80" } }, { "id": "electronic-carrier-statistics", "targets": [ { "kind": "node", "id": "CarrierMobility", "uid": "3303dee51d36b5a42bf80aa0ab62bdecd0db34cc77782c5203192ecf3512a91d" }, { "kind": "node", "id": "ElectronicDOS", "uid": "a32e08f3d5a147643986f961b701aea560791721df573f09188a557da369fb4f" }, { "kind": "operator", "id": "compute_carrier_mobility", "uid": "8de71440c5216dc390d9e5595d6e7dcc50a672fbc2c7c36400c04602cd1d2048", "formula_sha256": "21573bfd16d2a71a3fe127fb2685490564e562befd3119c79c02853ff268a121" }, { "kind": "operator", "id": "compute_electronic_dos", "uid": "ca35c6e4d21df105dfe83087ea98e03a5bad951006ada4fe722eea0a06a1cf7f", "formula_sha256": "b399aaa658591d443481b4cb89cc5eb171dee83c7744a60637547caacec53411" } ], "relation": "local", "statement": "compute_carrier_mobility is interpreted as amset's Boltzmann mobility mu = c sum_k w_k tau_k of one diagonal component, with c = e/n > 0, w_k >= 0 the Fermi-window and squared-velocity weights of the states k and tau_k = (sum_i 1/tau_{k,i})^-1 the state relaxation time of mechanism rates that add; mu_i = c sum_k w_k tau_{k,i} is the mobility with mechanism i alone. With every tau_{k,i} > 0, mu <= (sum_i 1/mu_i)^-1: Matthiessen's rule on the mechanism-resolved mobilities never underestimates the mobility. compute_electronic_dos is interpreted as the density of states sum_b g_b delta(E - E_b) of a finite set of levels with degeneracies g_b > 0, read against a test function. At k_B T > 0 the electron count N(mu) = sum_b g_b/(1 + exp((E_b - mu)/(k_B T))), the density of states integrated against the Fermi function, takes every value strictly between 0 and sum_b g_b at exactly one chemical potential.", "sources": [], "conditions": [ "compute_carrier_mobility is an opaque amset output in the catalog; amset computes mu = sigma/(n e) from the BoltzTraP2 transport integrals with its own mode-resolved rates. The interpretation covers one diagonal Cartesian component for one carrier type with nonnegative weights w_k (Fermi window times squared velocity times k-point weight, divided by n); off-diagonal components are not sign definite and are not covered.", "amset adds scattering rates in every state, 1/tau_k = sum_i 1/tau_{k,i}, and its separate_mobility output evaluates each mechanism's mobility with that mechanism's rates alone, which is mu_i here. The bound says that the overall mobility amset reports is at most the Matthiessen combination of its per-mechanism mobilities; the gap is not characterized, and equality is not claimed.", "compute_electronic_dos is an opaque Kohn-Sham output in the catalog. The interpretation is the delta sum of a finite set of levels E_b (the Kohn-Sham eigenvalues on a finite k mesh) with degeneracies g_b > 0 (spin times k-point weight), read against a test function; the Gaussian-smeared or tetrahedron histogram that the codes bin is not covered.", "The Fermi-level theorem needs k_B T > 0, at least one level and a count N with 0 < N < sum_b g_b (the number of states); at T = 0 every mu inside a gap gives the same count, so uniqueness fails there. The count of the delta-sum density of states, not of a smeared one, is what is inverted.", "Energies E_b, mu and k_B T in one unit (J, or eV with k_B in eV/K); tau in s; mu in m^2/(V s) with c = e/n in coherent SI (1 cm^2/(V s) = 1e-4 m^2/(V s)). The inequality and the count are unit-free beyond that." ], "conventions": { "model_scope": "Boltzmann mobility with state-by-state additive scattering rates, and Fermi-Dirac occupation of a finite level set. Physlib has no electronic scattering or density-of-states model, so these are local theorems over Mathlib reals that identify no Physlib declaration.", "component_scope": "One diagonal mobility component for one carrier type; one density of states of a finite level set.", "normalization_basis": "Mobility per carrier (sigma divided by n e); degeneracies g_b count states, so the electron count is per cell when the g_b are per cell.", "frequency_convention": "Not applicable; the catalog's phonon-frequency input to the mobility acts only through the scattering rates.", "unit_embedding": "Coherent SI for mu (m^2/(V s)) with c = e/n; energies, mu and k_B T in one unit for the count. Exact-real statements; floating-point evaluation is not certified.", "held_fixed": "The weights w_k, the mechanism relaxation times and c for the mobility; the levels, degeneracies and temperature for the count.", "admissibility": "c > 0, w_k >= 0 and every tau_{k,i} > 0 for the mobility bound; g_b > 0, k_B T > 0 and 0 < N < sum_b g_b for the Fermi level.", "reference_energy": "Levels and chemical potential share one energy origin; a common shift leaves the count unchanged.", "qpoint_weighting": "k-point weights are inside w_k and g_b; symmetry reduction and tetrahedron weights are not represented beyond that." }, "remaining_obligations": [ "The scattering-rate models (acoustic deformation potential, ionized impurity, piezoelectric, polar optical), amset's band interpolation and its carrier concentration n are not verified; the theorems hold for the interpreted sums.", "The size of the gap between the mobility and its Matthiessen combination, and the conditions for equality, are not characterized.", "The smeared density of states, k-point convergence, spin-orbit degeneracies and the carrier concentration as the count above the valence band are not covered." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIElectronicStatistics", "declarations": [ "OMAI.PhyslibBridge.ElectronicStatistics.boltzmannMobility_le_matthiessen", "OMAI.PhyslibBridge.ElectronicStatistics.exists_unique_fermiLevel" ], "source_path": "lean/bridge/OMAIElectronicStatistics.lean", "build_record": "lean/bridge/build-evidence.json", "source_sha256": "42647fea45250902515d99acdf1e8d2bf9d809ce1e15ff7ad0890fe59d05bfde" } }, { "id": "finite-displacement-force-constants", "relation": "local", "targets": [ { "kind": "operator", "id": "compute_fc2_finite_displacement", "uid": "8c4e324d9fb31b0cde5f98b83effdc03f0fe77d04b9caf332ceecff6fc16ece1", "formula_sha256": "ffda0ae6e61fea11db6294a25665260253cdb5d1af6e47e27ef07cd1bfba64e9" } ], "statement": "The central-difference estimate (F_j(u - h e_i) - F_j(u + h e_i))/(2h) of the catalog force constant -dF_j/du_i differs from it by at most M h^2 / 6, where M bounds the third derivative of F_j along the displacement on [-h, h]. The force must be differentiable at u and three times continuously differentiable along the displacement line, which holds for a four times continuously differentiable potential with F = -grad V.", "sources": [], "conditions": [ "The bound needs the third derivative of the force, the fourth derivative of the potential; a three times differentiable potential is not enough for an h^2 error.", "Central (plus-minus) displacements are assumed. When a site symmetry maps the plus displacement onto the minus one, phonopy omits the minus run and reconstructs its forces by symmetry, so the estimate is still the central difference. A truly one-sided difference (phonopy is_plusminus=False at a site without that symmetry) has an error of order h. Neither the symmetry argument nor the one-sided case is covered.", "The displacement step h > 0 is the same for the plus and minus configurations." ], "conventions": { "model_scope": "Finite-difference error of one force field along one displacement; a local theorem over Mathlib with no Physlib model.", "component_scope": "One displaced coordinate i and one force component j at a time.", "normalization_basis": "Force constants per pair of coordinates, not mass weighted.", "frequency_convention": "Not applicable; no frequencies enter these derivatives.", "unit_embedding": "Coherent SI: forces in N, displacements in m, force constants in N/m; phonopy's default step is 0.01 A (1e-12 m) in its VASP interface and 0.02 bohr (0.0106 A) in its QE interface.", "held_fixed": "All other coordinates stay at the reference configuration.", "admissibility": "h > 0, the force differentiable at u, and C^3 along the displacement with |d^3 F_j / du_i^3| <= M on [-h, h].", "reference_energy": "Not applicable; only forces enter.", "qpoint_weighting": "Not applicable; no Brillouin-zone sums enter these derivatives." }, "remaining_obligations": [ "Noise in the computed forces (SCF tolerance, k-point and smearing errors) adds an error of order (force noise)/h that the bound does not include.", "Supercell size and the folding of force constants of finite range are not verified." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIAtomisticDerivatives", "declarations": [ "OMAI.PhyslibBridge.Atomistic.centralDifference_error" ], "source_path": "lean/bridge/OMAIAtomisticDerivatives.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_fc2_finite_displacement" } ], "source_sha256": "c4b5049d77057a88a806c759d04159c752f5aeb8e30f2eb6ac10908c5a0e41a9" } }, { "id": "force-constant-sum-rules", "relation": "local", "targets": [ { "kind": "node", "id": "ForceConstants[order=2]", "uid": "78d005a949ff56a526a1911f0b24e766daecad0a7f5ce9c1cc15222f2ed73551" }, { "kind": "node", "id": "ForceConstants[order=3]", "uid": "73372380af126dde147e403f69339b87ee2f938a896dd5b56e9100c055dc8f93" }, { "kind": "operator", "id": "compute_force_constants[order=2]", "uid": "8ad53af4eefce437872dd08ee834962e6db76f7803ab50771b4985c0782c64d4", "formula_sha256": "2f30cf2ccf4f3b65e69e78385ddc551525d63f6f92ab6929cf5dfc387fc20dd9" }, { "kind": "operator", "id": "compute_force_constants[order=3]", "uid": "979756d34934118442ec22671b68e036b5e9751c95ed110d3a508ba2dee95288", "formula_sha256": "aa562fd454620e74d58b4d9cdb0abd8076dc039e13714849f5af3d972d25e7f8" } ], "statement": "For a potential of a finite set of sites invariant under rigid translations, whose first partial derivative dV/du_k is differentiable at the reference for the Phi2 rule and whose second-order force constant Phi2_{kl} is differentiable in the configuration there for the Phi3 rule (a potential twice or three times continuously differentiable at the reference qualifies), the catalog force constants satisfy the acoustic sum rules sum_s Phi2_{k,(s,beta)} = 0 and sum_s Phi3_{k,l,(s,gamma)} = 0 for every coordinate pair and direction. For a potential three times continuously differentiable at the reference, Phi3 is symmetric under every permutation of its three indices. A compiled two-atom potential has a nonzero Phi2, so the sum rule balances nonzero terms.", "sources": [], "conditions": [ "Sites form a finite set with every image explicit (a supercell or a cluster); the catalog's lattice labels R, R' become site labels. For an infinite crystal the statement applies to a supercell large enough that no interaction wraps onto itself.", "The second-order sum rule needs translation invariance and the first partial derivative dV/du_k differentiable at the reference configuration, the third-order rule the second-order force constant Phi2_{kl} differentiable in the configuration there; a potential twice or three times continuously differentiable at the reference satisfies these, and the permutation symmetry needs three times. Without these Mathlib's derivative is zero and the statements would hold only through that zero value." ], "conventions": { "model_scope": "Exact second and third derivatives of one potential of a finite set of sites; a local theorem over Mathlib with no Physlib model.", "component_scope": "All sites and Cartesian components; the sum is over the sites of one index at fixed Cartesian direction.", "normalization_basis": "Force constants per pair or triple of coordinates, not mass weighted.", "frequency_convention": "Not applicable; no frequencies enter these derivatives.", "unit_embedding": "Coherent SI: Phi2 in N/m (J/m^2), Phi3 in J/m^3; 1 eV/A^2 = 16.02176634 J/m^2.", "held_fixed": "The reference configuration u; no strain or temperature enters.", "admissibility": "Translation invariance at every configuration; the first partial derivative dV/du_k differentiable at the reference for the second-order rule, the second-order force constant Phi2_{kl} differentiable there for the third-order rule, the potential three times continuously differentiable there for the permutation symmetry.", "reference_energy": "An additive constant in V changes no force constant.", "qpoint_weighting": "Not applicable; no Brillouin-zone sums enter these derivatives." }, "remaining_obligations": [ "Numerical force constants violate the sum rules through finite differences, truncation and symmetrization; the corrections codes apply are not verified.", "Rotational (Born-Huang) invariance conditions, needed for the quadratic flexural branch of two-dimensional materials, are not covered.", "compute_gruneisen, which contracts Phi3 in the Maradudin-Fein form, is not covered." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIAtomisticDerivatives", "declarations": [ "OMAI.PhyslibBridge.Atomistic.forceConstant2_sum_eq_zero", "OMAI.PhyslibBridge.Atomistic.forceConstant3_sum_eq_zero", "OMAI.PhyslibBridge.Atomistic.forceConstant3_symm", "OMAI.PhyslibBridge.Atomistic.pairEnergy_forceConstant2", "OMAI.PhyslibBridge.Atomistic.pairEnergy_translate", "OMAI.PhyslibBridge.Atomistic.pairEnergy_contDiff" ], "source_path": "lean/bridge/OMAIAtomisticDerivatives.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_force_constants[order=2]" }, { "kind": "operator", "id": "compute_force_constants[order=3]" }, { "kind": "node", "id": "ForceConstants[order=2]" }, { "kind": "node", "id": "ForceConstants[order=3]" } ], "source_sha256": "c4b5049d77057a88a806c759d04159c752f5aeb8e30f2eb6ac10908c5a0e41a9" } }, { "id": "harmonic-joint-molar-ensemble", "targets": [ { "kind": "node", "id": "MolarEntropy", "uid": "0f08fcb37791dc7cc3e0e0ee5c611abbefc5075867719b275fcaa77ada4f1ac2" }, { "kind": "node", "id": "MolarHeatCapacity", "uid": "4fa2d8e16bde42f794f150f5d149e5d645d63f7ddaac59cef79bd41771b13c17" }, { "kind": "node", "id": "MolarHelmholtzFreeEnergy", "uid": "419335bd5f3a1949576e608e4147d8f18bef980a69a9c6b99199f812c806b7be" }, { "kind": "node", "id": "MolarInternalEnergy", "uid": "220926f4974baec5c511f5acae2c9c4e09a8276d41c1314391b91dec537ef2fb" }, { "kind": "operator", "id": "contract_molar_entropy", "uid": "c9322877acc6349a7e18b29646674344b12d39565adf4ee3e23f2b82e1b53ab3", "formula_sha256": "dd26ff6b4ba598e3847d7cb5bea43db12244846b2adeb279f4a3a9d3d0df168f" }, { "kind": "operator", "id": "contract_molar_free_energy", "uid": "9d4ded802829a8b2fe5544556eeb81534fe39e3ceae5efac0065e7ddcf3da152", "formula_sha256": "dbff2ebb2aece334d4cbf8a8ad1928d48e3660a69d64cb8b73746b54dedeb5b0" }, { "kind": "operator", "id": "contract_molar_heat_capacity", "uid": "00bbc32d579215c8ada01de36f6d5db89d0cd23ccd1e392bc4bc085e9c96fc8e", "formula_sha256": "d650d1a9d8bf0b675e7f416f15d6d29beda89179a7af59ab2afad6a1dd72e71c" }, { "kind": "operator", "id": "contract_molar_internal_energy", "uid": "77d1268cd072ac2acfbf95872ae6a8acc0a20ac9e196dfe7256048ad9f5f8f2d", "formula_sha256": "2df3e3d6b3c9796f181b1cd476cf85250ee59ca8d440c2a20770ef2bfe0d8065" } ], "relation": "conditional", "statement": "The full heterogeneous joint harmonic ensemble has the proved mode-product partition and additive F/U/S/C. Applying N_A/N_q to its actual thermodynamic quantities gives the four existing OMAI molar formulas.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.meanEnergy", "kind": "def", "line": 519, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L519" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.thermodynamicEntropy", "kind": "def", "line": 800, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L800" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.mathematicalPartitionFunction_add", "kind": "lemma", "line": 369, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L369" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.integrable_energy_add", "kind": "lemma", "line": 487, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L487" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.meanEnergy_add", "kind": "lemma", "line": 529, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L529" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.partitionFunction_add", "kind": "lemma", "line": 628, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L628" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy_add", "kind": "lemma", "line": 705, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L705" } ], "conditions": [ "Use a finite grid Fin N_q x Fin (3*N_atoms), with positive natural N_q and N_atoms. Fin labels map to OMAI 1-based labels by i.val+1.", "Each grid entry is an upstream positive-frequency harmonic oscillator with exact upstream hbar and kB; all entries share one positive absolute temperature.", "Use uniform q weights 1/N_q and a positive Avogadro scale N_A in the chosen molar units. Do not divide by the branch count or atom count.", "All modes must have strictly positive frequencies. An unmodified Gamma grid with zero acoustic modes does not satisfy the premises.", "Hold the mode spectrum, weights and primitive-cell basis fixed; convert stored linear THz to angular frequency.", "Construct the actual finite product of full Nat occupation spaces, with product counting measure, unweighted eigenvalue sum, dof=0 and phase-space unit=1. No occupation cutoff or equal-frequency assumption is used.", "Apply N_A/N_q after evaluating the joint thermodynamic quantities, not inside the Hamiltonian." ], "conventions": { "model_scope": "Finite heterogeneous independent-mode quantum harmonic product with all Nat occupations. The empty-index identity has one zero-energy microstate and partition one.", "component_scope": "Explicit measure-preserving rectangular-to-flat occupation reindexing for N_q by 3*N_atoms modes.", "normalization_basis": "N_A/N_q times the full mode sum, per mole of primitive cells, not per mole of atoms.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Per-mode energy in J and entropy/heat capacity in J/K; molar outputs in J/mol or J/(mol K) of primitive cells. The explicit positive Avogadro parameter is interpreted as N_A in the chosen units.", "held_fixed": "All frequencies, mode indices, q weights, primitive-cell basis and volume fixed in temperature derivatives.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "Uniform q weight 1/N_q, N_q>0; sum all 3*N_atoms branches." }, "remaining_obligations": [ "Intermode coupling, nonuniform q weights, zero or imaginary frequencies, phonon-mode production, thermodynamic limits and numerical implementations remain outside this independent-mode model." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicProduct", "declarations": [ "OMAI.PhyslibBridge.harmonicProduct_partitionFunction", "OMAI.PhyslibBridge.harmonicGridOccupation_measurePreserving", "OMAI.PhyslibBridge.harmonicGridEnsemble_energy", "OMAI.PhyslibBridge.harmonic_joint_molar_helmholtzFreeEnergy_eq_omai", "OMAI.PhyslibBridge.harmonic_joint_molar_meanEnergy_eq_omai", "OMAI.PhyslibBridge.harmonic_joint_molar_thermodynamicEntropy_eq_omai", "OMAI.PhyslibBridge.harmonic_joint_molar_heatCapacity_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicProduct.lean", "source_sha256": "4c0af6605e43ceed3460a8c7fe9198f486bd48cf2ac6e538fa0c45ac1a644578", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_molar_free_energy" }, { "kind": "node", "id": "MolarHelmholtzFreeEnergy" }, { "kind": "operator", "id": "contract_molar_internal_energy" }, { "kind": "node", "id": "MolarInternalEnergy" }, { "kind": "operator", "id": "contract_molar_entropy" }, { "kind": "node", "id": "MolarEntropy" }, { "kind": "operator", "id": "contract_molar_heat_capacity" }, { "kind": "node", "id": "MolarHeatCapacity" } ] } }, { "id": "harmonic-joint-volumetric-heat-capacity", "targets": [ { "kind": "node", "id": "VolumetricHeatCapacity", "uid": "29e324b97f31cdb7d3cd8ef6aec5668fff111fd7ee96bd19dd0426b8728c1d47" }, { "kind": "operator", "id": "contract_volumetric_heat_capacity", "uid": "239c7dd3370e7b7e4456f1ade35c5d81edba69efda786d25e8e1e74f6fa297d9", "formula_sha256": "e89d3db8b665f634538055816c2e1336259f88bcd007badcf318c802d2b3ff92" } ], "relation": "conditional", "statement": "The actual heterogeneous joint ensemble heat capacity divided by N_q*V_cell equals the same OMAI volumetric formula. This strengthens its model interpretation without adding another operator.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" } ], "conditions": [ "Use a finite grid Fin N_q x Fin (3*N_atoms), with positive natural N_q and N_atoms. Fin labels map to OMAI 1-based labels by i.val+1.", "Each grid entry is an upstream positive-frequency harmonic oscillator with exact upstream hbar and kB; all entries share one positive absolute temperature.", "All modes must have strictly positive frequencies. An unmodified Gamma grid with zero acoustic modes does not satisfy the premises.", "Hold the mode spectrum, weights and primitive-cell basis fixed; convert stored linear THz to angular frequency.", "V_cell is the positive primitive-cell volume in m^3, not a supercell volume; convert stored Angstrom^3 by the exact factor 10^-30.", "Use sum(c)/(N_q*V_cell), with no Avogadro factor or branch-count divisor. The molar comparison uses the same positive N_A and primitive-cell basis.", "Construct the actual finite product of full Nat occupation spaces, with product counting measure, unweighted eigenvalue sum, dof=0 and phase-space unit=1. No occupation cutoff or equal-frequency assumption is used.", "Apply 1/(N_q*V_cell) to the evaluated joint heat capacity, not inside the Hamiltonian." ], "conventions": { "model_scope": "Finite heterogeneous independent-mode quantum harmonic product with all Nat occupations. The empty-index identity has one zero-energy microstate and partition one.", "component_scope": "N_q q points and exactly 3*N_atoms branches per primitive-cell basis.", "normalization_basis": "Sum all 3*N_atoms branches; divide by N_q times primitive-cell volume, not by atom or branch count.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Per-mode heat capacity J/K and V_cell in m^3 give J/(m^3 K); Angstrom^3 converts exactly by 10^-30. Molar comparison uses J/(mol K) per mole of primitive cells.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "Each spectrum includes zero-point energy; its temperature derivative contributes no additive heat-capacity term.", "qpoint_weighting": "Uniform q weight 1/N_q, N_q>0; sum all 3*N_atoms branches." }, "remaining_obligations": [ "Primitive-cell geometry and phonon frequencies remain interpreted inputs, not verified producers. Nonuniform q, zero/imaginary modes, thermodynamic limits and finite-precision implementations remain outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibJointDensity", "declarations": [ "OMAI.PhyslibBridge.harmonic_joint_volumetric_heatCapacity_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibJointDensity.lean", "source_sha256": "9f5f729c3db288f9d5b7e1b285b050c40df8caeabf72df09591d6203c6e7fde7", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_volumetric_heat_capacity" }, { "kind": "node", "id": "VolumetricHeatCapacity" } ] } }, { "id": "harmonic-mode-entropy", "targets": [ { "kind": "node", "id": "Entropy", "uid": "5aa485c8eea2f4a2a9966f39589ad15e802327513f72c2366dc15605879b86ae" }, { "kind": "operator", "id": "compute_entropy", "uid": "40d04377570142198d0a7f3d505de3b8fa6fdcf0418622be679aceec11b13264", "formula_sha256": "a1368172bfefa61b76a34da3b474fa48f11a728e01b851f0abf1961cd53e61c7" } ], "relation": "conditional", "statement": "The complete Nat-spectrum harmonic ensemble's CanonicalEnsemble.thermodynamicEntropy equals OMAI's single-mode Entropy formula at positive temperature and angular frequency.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.thermodynamicEntropy", "kind": "def", "line": 800, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L800" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.helmholtzFreeEnergy_eq_meanEnergy_sub_temp_mul_thermodynamicEntropy", "kind": "theorem", "line": 173, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L173" } ], "conditions": [ "Use an upstream quantum harmonic oscillator with positive mass and angular frequency, exact upstream hbar and kB, and positive absolute temperature.", "Use the constructed Nat counting-measure ensemble with dof=0 and phase-space unit=1; the partition sum, Boltzmann finiteness and energy integrability are proved.", "Interpret Bose occupation as n_BE(x)=1/(exp(x)-1) with x=hbar*omega/(kB*T), and convert any stored linear THz to angular frequency." ], "conventions": { "model_scope": "Independent stable quantum harmonic mode at fixed volume, with countably infinite levels E_n=hbar*omega*(n+1/2).", "component_scope": "One fixed (q,nu) mode; no mixing of mode-indexed values and extensive ensemble values.", "normalization_basis": "One oscillator; counting measure, dof=0, phase-space unit=1.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Use the exact upstream hbar and kB in one consistent unit system: J, K and angular rad/s in SI; entropy and heat capacity are J/K.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "No q sum at the single-mode stage." }, "remaining_obligations": [ "The caller must justify this oscillator model and its frequency/unit interpretation for the material. No dynamical-matrix or phonon-producer theorem, zero-mode prescription or numerical implementation proof is supplied." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicEnsemble", "declarations": [ "OMAI.PhyslibBridge.harmonic_thermodynamicEntropy_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicEnsemble.lean", "source_sha256": "1a1f7e60f5cf4a3e468d173109b0127d7685123e6e0130b69374df687b94e172", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_entropy" }, { "kind": "node", "id": "Entropy" } ] } }, { "id": "harmonic-mode-free-energy", "targets": [ { "kind": "node", "id": "HelmholtzFreeEnergy", "uid": "38b082d42f5136a903752e5425c65c112c90bc564693ddb929ef3f175d9d96d8" }, { "kind": "operator", "id": "compute_free_energy", "uid": "b15ffc57501399eb9f1a0081d34ec1e16b579134c884d6dcf0b05434924e02eb", "formula_sha256": "919246645690d2a35e9330824756ff2cb9bae8fb4dfffa4b5f93b953a4ce178b" } ], "relation": "conditional", "statement": "The complete Nat-spectrum harmonic ensemble's CanonicalEnsemble.helmholtzFreeEnergy equals OMAI's single-mode HelmholtzFreeEnergy formula at positive temperature and angular frequency.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.helmholtzFreeEnergy_eq_meanEnergy_sub_temp_mul_thermodynamicEntropy", "kind": "theorem", "line": 173, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L173" } ], "conditions": [ "Use an upstream quantum harmonic oscillator with positive mass and angular frequency, exact upstream hbar and kB, and positive absolute temperature.", "Use the constructed Nat counting-measure ensemble with dof=0 and phase-space unit=1; the partition sum, Boltzmann finiteness and energy integrability are proved.", "Interpret Bose occupation as n_BE(x)=1/(exp(x)-1) with x=hbar*omega/(kB*T), and convert any stored linear THz to angular frequency." ], "conventions": { "model_scope": "Independent stable quantum harmonic mode at fixed volume, with countably infinite levels E_n=hbar*omega*(n+1/2).", "component_scope": "One fixed (q,nu) mode; no mixing of mode-indexed values and extensive ensemble values.", "normalization_basis": "One oscillator; counting measure, dof=0, phase-space unit=1.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Use the exact upstream hbar and kB in one consistent unit system: J, K and angular rad/s in SI; entropy and heat capacity are J/K.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "No q sum at the single-mode stage." }, "remaining_obligations": [ "The caller must justify this oscillator model and its frequency/unit interpretation for the material. No dynamical-matrix or phonon-producer theorem, zero-mode prescription or numerical implementation proof is supplied." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicEnsemble", "declarations": [ "OMAI.PhyslibBridge.harmonic_helmholtzFreeEnergy_eq_omai", "OMAI.PhyslibBridge.harmonicFreeEnergy_hasDerivAt" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicEnsemble.lean", "source_sha256": "1a1f7e60f5cf4a3e468d173109b0127d7685123e6e0130b69374df687b94e172", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_free_energy" }, { "kind": "node", "id": "HelmholtzFreeEnergy" } ] } }, { "id": "harmonic-mode-heat-capacity", "targets": [ { "kind": "node", "id": "HeatCapacity", "uid": "343adfdb0717f7f81ae52cc5fca8279ff84a9715ba19b5630d405453682b12f9" }, { "kind": "operator", "id": "compute_heat_capacity", "uid": "3662d6183de400b2337d314499e31dd473b7b2ea3994290f21046e36bdcedfdb", "formula_sha256": "b28a7bc9aae3f2172b22319465791fdb88fcfc5f7c0b8c3fcdfac99a9ded8bd9" } ], "relation": "conditional", "statement": "The complete Nat-spectrum harmonic ensemble's CanonicalEnsemble.heatCapacity equals OMAI's single-mode HeatCapacity formula at positive temperature and angular frequency.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" } ], "conditions": [ "Use an upstream quantum harmonic oscillator with positive mass and angular frequency, exact upstream hbar and kB, and positive absolute temperature.", "Use the constructed Nat counting-measure ensemble with dof=0 and phase-space unit=1; the partition sum, Boltzmann finiteness and energy integrability are proved.", "Interpret Bose occupation as n_BE(x)=1/(exp(x)-1) with x=hbar*omega/(kB*T), and convert any stored linear THz to angular frequency.", "Hold the spectrum fixed. The mean-energy equality on positive temperatures transfers its derivative to the upstream heat-capacity definition; no finite-state FDT or sum/derivative interchange is assumed." ], "conventions": { "model_scope": "Independent stable quantum harmonic mode at fixed volume, with countably infinite levels E_n=hbar*omega*(n+1/2).", "component_scope": "One fixed (q,nu) mode; no mixing of mode-indexed values and extensive ensemble values.", "normalization_basis": "One oscillator; counting measure, dof=0, phase-space unit=1.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Use the exact upstream hbar and kB in one consistent unit system: J, K and angular rad/s in SI; entropy and heat capacity are J/K.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "No q sum at the single-mode stage." }, "remaining_obligations": [ "The caller must justify this oscillator model and its frequency/unit interpretation for the material. No dynamical-matrix or phonon-producer theorem, zero-mode prescription or numerical implementation proof is supplied." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicEnsemble", "declarations": [ "OMAI.PhyslibBridge.harmonic_heatCapacity_eq_omai", "OMAI.PhyslibBridge.harmonicInternalEnergy_hasDerivAt", "OMAI.PhyslibBridge.harmonicHeatCapacity_eq_exp" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicEnsemble.lean", "source_sha256": "1a1f7e60f5cf4a3e468d173109b0127d7685123e6e0130b69374df687b94e172", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_heat_capacity" }, { "kind": "node", "id": "HeatCapacity" } ] } }, { "id": "harmonic-mode-internal-energy", "targets": [ { "kind": "node", "id": "InternalEnergy", "uid": "4e9c9eb989aa721167806f40b353dbb75aedba068a1a486eeef178f77432108d" }, { "kind": "operator", "id": "compute_internal_energy", "uid": "ce45fb09bf9c3f8076a2064b90853e163f80b20a3f5c7506af913c560accbc53", "formula_sha256": "6f0d545bd77043c0fa3b62ff5f97422a626b5ddd71b88baffe4d562b94d8f280" } ], "relation": "conditional", "statement": "The complete Nat-spectrum harmonic ensemble's CanonicalEnsemble.meanEnergy equals OMAI's single-mode InternalEnergy formula at positive temperature and angular frequency.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.meanEnergy", "kind": "def", "line": 519, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L519" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.helmholtzFreeEnergy_eq_meanEnergy_sub_temp_mul_thermodynamicEntropy", "kind": "theorem", "line": 173, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L173" } ], "conditions": [ "Use an upstream quantum harmonic oscillator with positive mass and angular frequency, exact upstream hbar and kB, and positive absolute temperature.", "Use the constructed Nat counting-measure ensemble with dof=0 and phase-space unit=1; the partition sum, Boltzmann finiteness and energy integrability are proved.", "Interpret Bose occupation as n_BE(x)=1/(exp(x)-1) with x=hbar*omega/(kB*T), and convert any stored linear THz to angular frequency." ], "conventions": { "model_scope": "Independent stable quantum harmonic mode at fixed volume, with countably infinite levels E_n=hbar*omega*(n+1/2).", "component_scope": "One fixed (q,nu) mode; no mixing of mode-indexed values and extensive ensemble values.", "normalization_basis": "One oscillator; counting measure, dof=0, phase-space unit=1.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Use the exact upstream hbar and kB in one consistent unit system: J, K and angular rad/s in SI; entropy and heat capacity are J/K.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "No q sum at the single-mode stage." }, "remaining_obligations": [ "The caller must justify this oscillator model and its frequency/unit interpretation for the material. No dynamical-matrix or phonon-producer theorem, zero-mode prescription or numerical implementation proof is supplied." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicEnsemble", "declarations": [ "OMAI.PhyslibBridge.harmonic_meanEnergy_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicEnsemble.lean", "source_sha256": "1a1f7e60f5cf4a3e468d173109b0127d7685123e6e0130b69374df687b94e172", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_internal_energy" }, { "kind": "node", "id": "InternalEnergy" } ] } }, { "id": "harmonic-molar-entropy", "targets": [ { "kind": "node", "id": "MolarEntropy", "uid": "0f08fcb37791dc7cc3e0e0ee5c611abbefc5075867719b275fcaa77ada4f1ac2" }, { "kind": "operator", "id": "contract_molar_entropy", "uid": "c9322877acc6349a7e18b29646674344b12d39565adf4ee3e23f2b82e1b53ab3", "formula_sha256": "dd26ff6b4ba598e3847d7cb5bea43db12244846b2adeb279f4a3a9d3d0df168f" } ], "relation": "conditional", "statement": "The uniform-q contraction N_A/N_q of the proved single-mode Physlib thermodynamicEntropy quantities equals OMAI's molar Entropy formula for a finite positive-frequency grid.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.thermodynamicEntropy", "kind": "def", "line": 800, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L800" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" } ], "conditions": [ "Use a finite grid Fin N_q x Fin (3*N_atoms), with positive natural N_q and N_atoms. Fin labels map to OMAI 1-based labels by i.val+1.", "Each grid entry is an upstream positive-frequency harmonic oscillator with exact upstream hbar and kB; all entries share one positive absolute temperature.", "Use uniform q weights 1/N_q and a positive Avogadro scale N_A in the chosen molar units. Do not divide by the branch count or atom count.", "All modes must have strictly positive frequencies. An unmodified Gamma grid with zero acoustic modes does not satisfy the premises.", "Hold the mode spectrum, weights and primitive-cell basis fixed; convert stored linear THz to angular frequency." ], "conventions": { "model_scope": "Normalized sum of proved single-mode ensemble quantities; the separate harmonic-joint-molar-ensemble record establishes the joint-product interpretation.", "component_scope": "N_q q points and exactly 3*N_atoms branches per primitive-cell basis.", "normalization_basis": "N_A/N_q times the full mode sum, per mole of primitive cells, not per mole of atoms.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Per-mode energy in J and entropy/heat capacity in J/K; molar outputs in J/mol or J/(mol K) of primitive cells. The explicit positive Avogadro parameter is interpreted as N_A in the chosen units.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "Uniform q weight 1/N_q, N_q>0; sum all 3*N_atoms branches." }, "remaining_obligations": [ "This record retains the single-mode contraction proof. The separate harmonic-joint-molar-ensemble record supplies joint-product thermodynamics. Nonuniform q, zero or imaginary modes, phonon producers, thermodynamic limits and numerical implementations remain outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicSums", "declarations": [ "OMAI.PhyslibBridge.harmonic_molar_thermodynamicEntropy_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicSums.lean", "source_sha256": "5b3c3afbfc6a0fe5e9951091d66fc61b0802ebd6b5ca786927c67ecba2193e52", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_molar_entropy" }, { "kind": "node", "id": "MolarEntropy" } ] } }, { "id": "harmonic-molar-free-energy", "targets": [ { "kind": "node", "id": "MolarHelmholtzFreeEnergy", "uid": "419335bd5f3a1949576e608e4147d8f18bef980a69a9c6b99199f812c806b7be" }, { "kind": "operator", "id": "contract_molar_free_energy", "uid": "9d4ded802829a8b2fe5544556eeb81534fe39e3ceae5efac0065e7ddcf3da152", "formula_sha256": "dbff2ebb2aece334d4cbf8a8ad1928d48e3660a69d64cb8b73746b54dedeb5b0" } ], "relation": "conditional", "statement": "The uniform-q contraction N_A/N_q of the proved single-mode Physlib helmholtzFreeEnergy quantities equals OMAI's molar HelmholtzFreeEnergy formula for a finite positive-frequency grid.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" } ], "conditions": [ "Use a finite grid Fin N_q x Fin (3*N_atoms), with positive natural N_q and N_atoms. Fin labels map to OMAI 1-based labels by i.val+1.", "Each grid entry is an upstream positive-frequency harmonic oscillator with exact upstream hbar and kB; all entries share one positive absolute temperature.", "Use uniform q weights 1/N_q and a positive Avogadro scale N_A in the chosen molar units. Do not divide by the branch count or atom count.", "All modes must have strictly positive frequencies. An unmodified Gamma grid with zero acoustic modes does not satisfy the premises.", "Hold the mode spectrum, weights and primitive-cell basis fixed; convert stored linear THz to angular frequency." ], "conventions": { "model_scope": "Normalized sum of proved single-mode ensemble quantities; the separate harmonic-joint-molar-ensemble record establishes the joint-product interpretation.", "component_scope": "N_q q points and exactly 3*N_atoms branches per primitive-cell basis.", "normalization_basis": "N_A/N_q times the full mode sum, per mole of primitive cells, not per mole of atoms.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Per-mode energy in J and entropy/heat capacity in J/K; molar outputs in J/mol or J/(mol K) of primitive cells. The explicit positive Avogadro parameter is interpreted as N_A in the chosen units.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "Uniform q weight 1/N_q, N_q>0; sum all 3*N_atoms branches." }, "remaining_obligations": [ "This record retains the single-mode contraction proof. The separate harmonic-joint-molar-ensemble record supplies joint-product thermodynamics. Nonuniform q, zero or imaginary modes, phonon producers, thermodynamic limits and numerical implementations remain outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicSums", "declarations": [ "OMAI.PhyslibBridge.harmonic_molar_helmholtzFreeEnergy_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicSums.lean", "source_sha256": "5b3c3afbfc6a0fe5e9951091d66fc61b0802ebd6b5ca786927c67ecba2193e52", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_molar_free_energy" }, { "kind": "node", "id": "MolarHelmholtzFreeEnergy" } ] } }, { "id": "harmonic-molar-heat-capacity", "targets": [ { "kind": "node", "id": "MolarHeatCapacity", "uid": "4fa2d8e16bde42f794f150f5d149e5d645d63f7ddaac59cef79bd41771b13c17" }, { "kind": "operator", "id": "contract_molar_heat_capacity", "uid": "00bbc32d579215c8ada01de36f6d5db89d0cd23ccd1e392bc4bc085e9c96fc8e", "formula_sha256": "d650d1a9d8bf0b675e7f416f15d6d29beda89179a7af59ab2afad6a1dd72e71c" } ], "relation": "conditional", "statement": "The uniform-q contraction N_A/N_q of the proved single-mode Physlib heatCapacity quantities equals OMAI's molar HeatCapacity formula for a finite positive-frequency grid.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" } ], "conditions": [ "Use a finite grid Fin N_q x Fin (3*N_atoms), with positive natural N_q and N_atoms. Fin labels map to OMAI 1-based labels by i.val+1.", "Each grid entry is an upstream positive-frequency harmonic oscillator with exact upstream hbar and kB; all entries share one positive absolute temperature.", "Use uniform q weights 1/N_q and a positive Avogadro scale N_A in the chosen molar units. Do not divide by the branch count or atom count.", "All modes must have strictly positive frequencies. An unmodified Gamma grid with zero acoustic modes does not satisfy the premises.", "Hold the mode spectrum, weights and primitive-cell basis fixed; convert stored linear THz to angular frequency." ], "conventions": { "model_scope": "Normalized sum of proved single-mode ensemble quantities; the separate harmonic-joint-molar-ensemble record establishes the joint-product interpretation.", "component_scope": "N_q q points and exactly 3*N_atoms branches per primitive-cell basis.", "normalization_basis": "N_A/N_q times the full mode sum, per mole of primitive cells, not per mole of atoms.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Per-mode energy in J and entropy/heat capacity in J/K; molar outputs in J/mol or J/(mol K) of primitive cells. The explicit positive Avogadro parameter is interpreted as N_A in the chosen units.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "Uniform q weight 1/N_q, N_q>0; sum all 3*N_atoms branches." }, "remaining_obligations": [ "This record retains the single-mode contraction proof. The separate harmonic-joint-molar-ensemble record supplies joint-product thermodynamics. Nonuniform q, zero or imaginary modes, phonon producers, thermodynamic limits and numerical implementations remain outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicSums", "declarations": [ "OMAI.PhyslibBridge.harmonic_molar_heatCapacity_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicSums.lean", "source_sha256": "5b3c3afbfc6a0fe5e9951091d66fc61b0802ebd6b5ca786927c67ecba2193e52", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_molar_heat_capacity" }, { "kind": "node", "id": "MolarHeatCapacity" } ] } }, { "id": "harmonic-molar-internal-energy", "targets": [ { "kind": "node", "id": "MolarInternalEnergy", "uid": "220926f4974baec5c511f5acae2c9c4e09a8276d41c1314391b91dec537ef2fb" }, { "kind": "operator", "id": "contract_molar_internal_energy", "uid": "77d1268cd072ac2acfbf95872ae6a8acc0a20ac9e196dfe7256048ad9f5f8f2d", "formula_sha256": "2df3e3d6b3c9796f181b1cd476cf85250ee59ca8d440c2a20770ef2bfe0d8065" } ], "relation": "conditional", "statement": "The uniform-q contraction N_A/N_q of the proved single-mode Physlib meanEnergy quantities equals OMAI's molar InternalEnergy formula for a finite positive-frequency grid.", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.meanEnergy", "kind": "def", "line": 519, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L519" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" } ], "conditions": [ "Use a finite grid Fin N_q x Fin (3*N_atoms), with positive natural N_q and N_atoms. Fin labels map to OMAI 1-based labels by i.val+1.", "Each grid entry is an upstream positive-frequency harmonic oscillator with exact upstream hbar and kB; all entries share one positive absolute temperature.", "Use uniform q weights 1/N_q and a positive Avogadro scale N_A in the chosen molar units. Do not divide by the branch count or atom count.", "All modes must have strictly positive frequencies. An unmodified Gamma grid with zero acoustic modes does not satisfy the premises.", "Hold the mode spectrum, weights and primitive-cell basis fixed; convert stored linear THz to angular frequency." ], "conventions": { "model_scope": "Normalized sum of proved single-mode ensemble quantities; the separate harmonic-joint-molar-ensemble record establishes the joint-product interpretation.", "component_scope": "N_q q points and exactly 3*N_atoms branches per primitive-cell basis.", "normalization_basis": "N_A/N_q times the full mode sum, per mole of primitive cells, not per mole of atoms.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Per-mode energy in J and entropy/heat capacity in J/K; molar outputs in J/mol or J/(mol K) of primitive cells. The explicit positive Avogadro parameter is interpreted as N_A in the chosen units.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "The spectrum includes hbar*omega/2; this constant contributes to F and U but not to S or C.", "qpoint_weighting": "Uniform q weight 1/N_q, N_q>0; sum all 3*N_atoms branches." }, "remaining_obligations": [ "This record retains the single-mode contraction proof. The separate harmonic-joint-molar-ensemble record supplies joint-product thermodynamics. Nonuniform q, zero or imaginary modes, phonon producers, thermodynamic limits and numerical implementations remain outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicSums", "declarations": [ "OMAI.PhyslibBridge.harmonic_molar_meanEnergy_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicSums.lean", "source_sha256": "5b3c3afbfc6a0fe5e9951091d66fc61b0802ebd6b5ca786927c67ecba2193e52", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_molar_internal_energy" }, { "kind": "node", "id": "MolarInternalEnergy" } ] } }, { "id": "harmonic-qha-bulk-modulus", "relation": "conditional", "statement": "For the same family, the isothermal curvature modulus V*(d2F/dV2)_T at the volume (dG/dp)_T selected by the windowed Gibbs energy equals -V/(dV/dp)_T read off the Gibbs energy alone. Its positivity is the stability premise F_VV>0, not a separate result.", "targets": [ { "kind": "node", "id": "BulkModulus", "uid": "c0a13e64c02811f49b951271c894373267937676e0c657fc9ef09fac72558ced" }, { "kind": "operator", "id": "compute_bulk_modulus_qha", "uid": "de67212047d6c2bb08a94d9394adffe1b1cc183b7a1811d838de2dcfb9323235", "formula_sha256": "bedbb8f82d002302668176dcdb9aa7abcbbe648e5963121237de436c725b2e7b" } ], "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" } ], "conditions": [ "Use the same finite family of full Nat harmonic occupation ensembles as the Cp binding: product counting measure, unweighted oscillator energies, dof=0 and phase-space unit=1, with static energy N_q*E_0(v) attached through an actual singleton ensemble before molar normalization.", "Temperature T and primitive-cell volume v are positive. Every frequency is positive and C2 on an open positive-volume domain. Static energy E_0(v) is C2, arbitrary signed, and independent of temperature. No mode may have zero or imaginary frequency; an unmodified Gamma-centred grid with zero acoustic modes does not satisfy the premises.", "Keep masses, composition, atom and q counts, reduced-q labels, uniform q weights and the Avogadro scale fixed, with one q mesh at every volume. Frequencies depend on volume only.", "The volume family V(T,p) is differentiable at every point of a neighborhood of the evaluation point (T,p), stays in the model domain and satisfies F_V(T,V)+p=0 throughout that neighborhood, with F_VV>0 at the evaluation point. The implicit-function theorem supplies such a family through every stable stationary point; a pointwise stationarity equation alone is insufficient.", "The Gibbs energy is the infimum of F(T,V)+pV over an open volume window (a,b) that contains the selected molar volume and lies in the model domain. The window selects the family when F_VV(T,V)>0 for every V in (a,b) and every T near the evaluation temperature: a scan certifies itself by this convexity of F+pV across the scanned interval. The selected volume must lie strictly inside the scanned range; EOS extrapolation beyond the scanned volumes is not covered. Every sufficiently narrow window around the selected volume also selects the family; a window that reaches another local minimum can select another branch, which is not covered.", "Interpret K^qha(E_tot,S,omega) as the isothermal modulus of the actual molar Helmholtz surface at the selected volume and temperature, the faithful mirror of the Phonopy EOS B0. The adiabatic modulus K_S=K_T*C_P/C_V and static-only curvature are different quantities and are not this output.", "The modulus is taken along the model's single hydrostatic volume coordinate. A fixed-shape lattice scan of a non-cubic crystal gives a fixed-shape modulus and a shape-relaxed path a Reuss-type modulus. The BulkModulus node is defined as the Voigt average of the elastic tensor; it is listed as the output association of this binding and is not a proof subject.", "Embed model values coherently in SI: T in K, p in Pa, E_0 in J per primitive cell, molar F and G in J/mol of primitive cells, v in m^3, V_m in m^3/mol, alpha_V in 1/K, K_T in Pa and capacities in J/(mol K) or J/(m^3 K). Convert Angstrom^3 by 10^-30, serving GPa by 10^9 and eV by 1.602176634e-19 J/eV. These exact-real interpretations do not certify floating-point adapters." ], "conventions": { "model_scope": "Local hydrostatic static-plus-independent-harmonic canonical model with volume-dependent positive frequencies and no independent temperature dependence of the spectrum. The Gibbs energy is a windowed minimum over molar volume of the actual molar Helmholtz surface plus pV. This is an OMAI model-specific extension of actual Physlib ensembles, not an upstream QHA theorem or a numerical QHA solver proof.", "component_scope": "Finite uniform-q grid with all 3*N_atoms branches and fixed reduced-q labels; one scalar volume coordinate along a hydrostatic path, one temperature and one pressure coordinate. Electronic, magnetic, configurational, anharmonic and anisotropic strain contributions are excluded.", "normalization_basis": "Moles of primitive cells with the Avogadro scale N_A=6.02214076e23 mol^-1 for the J/mol embedding. F, G and C_P are molar in that basis and scale with N_A; C_V_vol divides by the primitive cell volume. alpha_V, K_T and Gamma are intensive and do not depend on N_A.", "frequency_convention": "Angular SI frequency with Physlib's CODATA hbar 1.054571817e-34 J s and exact kB in the ensemble energies and heat-capacity weights. Stored linear THz requires omega_SI=2*pi*10^12*nu_THz.", "unit_embedding": "G and F in J/mol, pressure and K_T in Pa, V_m in m^3/mol, alpha_V in 1/K, C_P in J/(mol K) and C_V_vol in J/(m^3 K). Use a common numeric-unit embedding, not raw mixed serving values.", "held_fixed": "Composition, masses, q labels and weights, the q mesh, the Avogadro scale and the static-energy and frequency functions are fixed. C_P and alpha_V differentiate at constant pressure, K_T at constant temperature, C_V at constant volume.", "admissibility": "T>0, v>0, omega>0, N_A>0, N_q>0, local positive isothermal curvature and a window inside the locally convex region. alpha_V and Gamma are sign-free. Totalized values outside the model domain or at T=0 have no asserted physical interpretation.", "reference_energy": "All oscillator zero-point energies remain and shift the selected volume and modulus. The signed temperature-independent static energy enters the actual Hamiltonian and is not silently removed.", "qpoint_weighting": "Uniform full-grid q averaging with the same mesh at every volume; varying primitive volume does not change reduced-q labels or introduce fractional oscillator copies." }, "remaining_obligations": [ "Static-energy, phonon and geometry producers are interpreted inputs, not verified implementations. Numerical volume scans, EOS fits, polynomial or finite-difference derivatives, minimizers and the Phonopy QHA code are not verified; the theorems state what such a producer computes on an exact model, not that a given code does.", "Global minimization over all volumes, first-order transitions between branches, phase coexistence, mechanically unstable branches and the T=0 limit are outside scope.", "Zero or imaginary modes, Gamma-point acoustic modes, nonuniform or symmetry-reduced q quadrature, q meshes that change with volume, independent temperature-dependent spectra, anharmonic, electronic or magnetic free energies and anisotropic strain remain outside scope.", "A producer must reject any imaginary mode at every scanned volume, not only at the selected one. Gamma-point acoustic modes are excluded; a shifted mesh without the Gamma point meets the premises. For polar materials the Born charges and eps_inf of the nonanalytic correction must be recomputed at every volume. The static energy E(V) and the phonon free energy must refer to the same cell at every volume.", "The theorems hold for any T>0 with F_VV>0, including temperatures where the quasiharmonic approximation is physically unreliable, such as strong anharmonicity near melting or soft-mode transitions. They are statements about the model, not about its adequacy.", "Compiled witnesses establish realizable model examples with positive, negative and zero expansion; they do not certify any material or numerical volume scan." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibQHA", "declarations": [ "OMAI.PhyslibBridge.Response.harmonic_static_qhaBulkModulus_eq_omai", "OMAI.PhyslibBridge.HarmonicVolumeModel.freeEnergy_eq_actual" ], "source_path": "lean/bridge/OMAIPhyslibQHA.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_bulk_modulus_qha" } ], "source_sha256": "2347e44452e37e3a05131ad636dd20f6ad7ea74dcc60c97fb8a2134f63f68f96" } }, { "id": "harmonic-qha-gibbs-energy", "relation": "conditional", "statement": "For the static-plus-harmonic Physlib ensemble, the infimum of F(T,V)+pV over every sufficiently narrow volume window around a local stable equilibrium family is attained uniquely at the family volume, and so is the infimum over any open window across which F+pV is convex. Its temperature derivative is -(N_A/N_q) times the Physlib thermodynamic entropy of the ensemble at the selected volume, and its pressure derivative is the selected molar volume.", "targets": [ { "kind": "node", "id": "QHAGibbsEnergy", "uid": "bb10912780eb22d45beff551be14f7f4216f3aae26161e8bfd60cdc051f63f58" }, { "kind": "operator", "id": "compute_qha_gibbs", "uid": "2dfd344c598d659dbc5e4c8def4cefcb69b1a0bd215a9202f981618b18611b89", "formula_sha256": "8264d933a1f5cf6dab090041c2a1e6367cd92e5de6022aaa8e7dbbbe20ed840e" } ], "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.thermodynamicEntropy", "kind": "def", "line": 800, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L800" } ], "conditions": [ "Use the same finite family of full Nat harmonic occupation ensembles as the Cp binding: product counting measure, unweighted oscillator energies, dof=0 and phase-space unit=1, with static energy N_q*E_0(v) attached through an actual singleton ensemble before molar normalization.", "Temperature T and primitive-cell volume v are positive. Every frequency is positive and C2 on an open positive-volume domain. Static energy E_0(v) is C2, arbitrary signed, and independent of temperature. No mode may have zero or imaginary frequency; an unmodified Gamma-centred grid with zero acoustic modes does not satisfy the premises.", "Keep masses, composition, atom and q counts, reduced-q labels, uniform q weights and the Avogadro scale fixed, with one q mesh at every volume. Frequencies depend on volume only.", "The volume family V(T,p) is differentiable at every point of a neighborhood of the evaluation point (T,p), stays in the model domain and satisfies F_V(T,V)+p=0 throughout that neighborhood, with F_VV>0 at the evaluation point. The implicit-function theorem supplies such a family through every stable stationary point; a pointwise stationarity equation alone is insufficient.", "The Gibbs energy is the infimum of F(T,V)+pV over an open volume window (a,b) that contains the selected molar volume and lies in the model domain. The window selects the family when F_VV(T,V)>0 for every V in (a,b) and every T near the evaluation temperature: a scan certifies itself by this convexity of F+pV across the scanned interval. The selected volume must lie strictly inside the scanned range; EOS extrapolation beyond the scanned volumes is not covered. Every sufficiently narrow window around the selected volume also selects the family; a window that reaches another local minimum can select another branch, which is not covered.", "Interpret G^qha(E_tot,S,omega) as that windowed infimum with F the actual molar Helmholtz surface built from the static energy, structure and frequencies of one primitive cell. The entropy term is -(N_A/N_q) times the Physlib thermodynamic entropy of the static-plus-harmonic ensemble at the selected volume.", "Embed model values coherently in SI: T in K, p in Pa, E_0 in J per primitive cell, molar F and G in J/mol of primitive cells, v in m^3, V_m in m^3/mol, alpha_V in 1/K, K_T in Pa and capacities in J/(mol K) or J/(m^3 K). Convert Angstrom^3 by 10^-30, serving GPa by 10^9 and eV by 1.602176634e-19 J/eV. These exact-real interpretations do not certify floating-point adapters." ], "conventions": { "model_scope": "Local hydrostatic static-plus-independent-harmonic canonical model with volume-dependent positive frequencies and no independent temperature dependence of the spectrum. The Gibbs energy is a windowed minimum over molar volume of the actual molar Helmholtz surface plus pV. This is an OMAI model-specific extension of actual Physlib ensembles, not an upstream QHA theorem or a numerical QHA solver proof.", "component_scope": "Finite uniform-q grid with all 3*N_atoms branches and fixed reduced-q labels; one scalar volume coordinate along a hydrostatic path, one temperature and one pressure coordinate. Electronic, magnetic, configurational, anharmonic and anisotropic strain contributions are excluded.", "normalization_basis": "Moles of primitive cells with the Avogadro scale N_A=6.02214076e23 mol^-1 for the J/mol embedding. F, G and C_P are molar in that basis and scale with N_A; C_V_vol divides by the primitive cell volume. alpha_V, K_T and Gamma are intensive and do not depend on N_A.", "frequency_convention": "Angular SI frequency with Physlib's CODATA hbar 1.054571817e-34 J s and exact kB in the ensemble energies and heat-capacity weights. Stored linear THz requires omega_SI=2*pi*10^12*nu_THz.", "unit_embedding": "G and F in J/mol, pressure and K_T in Pa, V_m in m^3/mol, alpha_V in 1/K, C_P in J/(mol K) and C_V_vol in J/(m^3 K). Use a common numeric-unit embedding, not raw mixed serving values.", "held_fixed": "Composition, masses, q labels and weights, the q mesh, the Avogadro scale and the static-energy and frequency functions are fixed. C_P and alpha_V differentiate at constant pressure, K_T at constant temperature, C_V at constant volume.", "admissibility": "T>0, v>0, omega>0, N_A>0, N_q>0, local positive isothermal curvature and a window inside the locally convex region. alpha_V and Gamma are sign-free. Totalized values outside the model domain or at T=0 have no asserted physical interpretation.", "reference_energy": "All oscillator zero-point energies remain and shift the selected volume and modulus. The signed temperature-independent static energy enters the actual Hamiltonian and is not silently removed.", "qpoint_weighting": "Uniform full-grid q averaging with the same mesh at every volume; varying primitive volume does not change reduced-q labels or introduce fractional oscillator copies." }, "remaining_obligations": [ "Static-energy, phonon and geometry producers are interpreted inputs, not verified implementations. Numerical volume scans, EOS fits, polynomial or finite-difference derivatives, minimizers and the Phonopy QHA code are not verified; the theorems state what such a producer computes on an exact model, not that a given code does.", "Global minimization over all volumes, first-order transitions between branches, phase coexistence, mechanically unstable branches and the T=0 limit are outside scope.", "Zero or imaginary modes, Gamma-point acoustic modes, nonuniform or symmetry-reduced q quadrature, q meshes that change with volume, independent temperature-dependent spectra, anharmonic, electronic or magnetic free energies and anisotropic strain remain outside scope.", "A producer must reject any imaginary mode at every scanned volume, not only at the selected one. Gamma-point acoustic modes are excluded; a shifted mesh without the Gamma point meets the premises. For polar materials the Born charges and eps_inf of the nonanalytic correction must be recomputed at every volume. The static energy E(V) and the phonon free energy must refer to the same cell at every volume.", "The theorems hold for any T>0 with F_VV>0, including temperatures where the quasiharmonic approximation is physically unreliable, such as strong anharmonicity near melting or soft-mode transitions. They are statements about the model, not about its adequacy.", "Compiled witnesses establish realizable model examples with positive, negative and zero expansion; they do not certify any material or numerical volume scan." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibQHA", "declarations": [ "OMAI.PhyslibBridge.Response.harmonic_static_qhaGibbs_convex_window", "OMAI.PhyslibBridge.Response.harmonic_static_qhaGibbs_entropy", "OMAI.PhyslibBridge.Response.harmonic_static_qhaGibbs_volume" ], "source_path": "lean/bridge/OMAIPhyslibQHA.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_qha_gibbs" }, { "kind": "node", "id": "QHAGibbsEnergy" } ], "source_sha256": "2347e44452e37e3a05131ad636dd20f6ad7ea74dcc60c97fb8a2134f63f68f96" } }, { "id": "harmonic-qha-heat-capacity-p", "relation": "conditional", "statement": "For the same windowed Gibbs energy, -T times its second temperature derivative at constant pressure equals the temperature derivative of actual molar enthalpy along the constant-pressure path, equals C_V+T*V*alpha_V^2*K_T with alpha_V and K_T read off the same Gibbs energy, and equals C_V+T*(dV/dT)_p*(dS/dV)_T with the Maxwell relation (dS/dV)_T=-F_TV=alpha_V*K_T, the expression evaluated by the catalog default scheme polyfit_enthalpy_derivative. It exceeds C_V exactly when alpha_V is nonzero.", "targets": [ { "kind": "node", "id": "HeatCapacityConstantP", "uid": "08057e4554a3420c933329b3821de9295e2b5950191e23cafe310b0a15adc7b6" }, { "kind": "node", "id": "QHAGibbsEnergy", "uid": "bb10912780eb22d45beff551be14f7f4216f3aae26161e8bfd60cdc051f63f58" }, { "kind": "operator", "id": "compute_heat_capacity_p", "uid": "36100a86ef25aada0acd615151092323ba680b4a9791bb58d1175adcea96deb2", "formula_sha256": "94c543b5191d7fa6737a2f2fcca9cc68bc4cc38530abd5c2d31b25b091bc4ccf" } ], "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.meanEnergyT", "kind": "def", "line": 409, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L409" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" } ], "conditions": [ "Use the same finite family of full Nat harmonic occupation ensembles as the Cp binding: product counting measure, unweighted oscillator energies, dof=0 and phase-space unit=1, with static energy N_q*E_0(v) attached through an actual singleton ensemble before molar normalization.", "Temperature T and primitive-cell volume v are positive. Every frequency is positive and C2 on an open positive-volume domain. Static energy E_0(v) is C2, arbitrary signed, and independent of temperature. No mode may have zero or imaginary frequency; an unmodified Gamma-centred grid with zero acoustic modes does not satisfy the premises.", "Keep masses, composition, atom and q counts, reduced-q labels, uniform q weights and the Avogadro scale fixed, with one q mesh at every volume. Frequencies depend on volume only.", "The volume family V(T,p) is differentiable at every point of a neighborhood of the evaluation point (T,p), stays in the model domain and satisfies F_V(T,V)+p=0 throughout that neighborhood, with F_VV>0 at the evaluation point. The implicit-function theorem supplies such a family through every stable stationary point; a pointwise stationarity equation alone is insufficient.", "The Gibbs energy is the infimum of F(T,V)+pV over an open volume window (a,b) that contains the selected molar volume and lies in the model domain. The window selects the family when F_VV(T,V)>0 for every V in (a,b) and every T near the evaluation temperature: a scan certifies itself by this convexity of F+pV across the scanned interval. The selected volume must lie strictly inside the scanned range; EOS extrapolation beyond the scanned volumes is not covered. Every sufficiently narrow window around the selected volume also selects the family; a window that reaches another local minimum can select another branch, which is not covered.", "Interpret C_P^qha(G_qha) as -T*(d2G/dT2)_p of that windowed Gibbs energy at the evaluation point. C_V is the fixed-volume capacity of the same actual ensemble at the selected volume and S its molar entropy; alpha_V=(dV/dT)_p/V and K_T are the producers of the sibling bindings on the same Gibbs energy.", "Embed model values coherently in SI: T in K, p in Pa, E_0 in J per primitive cell, molar F and G in J/mol of primitive cells, v in m^3, V_m in m^3/mol, alpha_V in 1/K, K_T in Pa and capacities in J/(mol K) or J/(m^3 K). Convert Angstrom^3 by 10^-30, serving GPa by 10^9 and eV by 1.602176634e-19 J/eV. These exact-real interpretations do not certify floating-point adapters." ], "conventions": { "model_scope": "Local hydrostatic static-plus-independent-harmonic canonical model with volume-dependent positive frequencies and no independent temperature dependence of the spectrum. The Gibbs energy is a windowed minimum over molar volume of the actual molar Helmholtz surface plus pV. This is an OMAI model-specific extension of actual Physlib ensembles, not an upstream QHA theorem or a numerical QHA solver proof.", "component_scope": "Finite uniform-q grid with all 3*N_atoms branches and fixed reduced-q labels; one scalar volume coordinate along a hydrostatic path, one temperature and one pressure coordinate. Electronic, magnetic, configurational, anharmonic and anisotropic strain contributions are excluded.", "normalization_basis": "Moles of primitive cells with the Avogadro scale N_A=6.02214076e23 mol^-1 for the J/mol embedding. F, G and C_P are molar in that basis and scale with N_A; C_V_vol divides by the primitive cell volume. alpha_V, K_T and Gamma are intensive and do not depend on N_A.", "frequency_convention": "Angular SI frequency with Physlib's CODATA hbar 1.054571817e-34 J s and exact kB in the ensemble energies and heat-capacity weights. Stored linear THz requires omega_SI=2*pi*10^12*nu_THz.", "unit_embedding": "G and F in J/mol, pressure and K_T in Pa, V_m in m^3/mol, alpha_V in 1/K, C_P in J/(mol K) and C_V_vol in J/(m^3 K). Use a common numeric-unit embedding, not raw mixed serving values.", "held_fixed": "Composition, masses, q labels and weights, the q mesh, the Avogadro scale and the static-energy and frequency functions are fixed. C_P and alpha_V differentiate at constant pressure, K_T at constant temperature, C_V at constant volume.", "admissibility": "T>0, v>0, omega>0, N_A>0, N_q>0, local positive isothermal curvature and a window inside the locally convex region. alpha_V and Gamma are sign-free. Totalized values outside the model domain or at T=0 have no asserted physical interpretation.", "reference_energy": "All oscillator zero-point energies remain and shift the selected volume and modulus. The signed temperature-independent static energy enters the actual Hamiltonian and is not silently removed.", "qpoint_weighting": "Uniform full-grid q averaging with the same mesh at every volume; varying primitive volume does not change reduced-q labels or introduce fractional oscillator copies." }, "remaining_obligations": [ "Static-energy, phonon and geometry producers are interpreted inputs, not verified implementations. Numerical volume scans, EOS fits, polynomial or finite-difference derivatives, minimizers and the Phonopy QHA code are not verified; the theorems state what such a producer computes on an exact model, not that a given code does.", "Global minimization over all volumes, first-order transitions between branches, phase coexistence, mechanically unstable branches and the T=0 limit are outside scope.", "Zero or imaginary modes, Gamma-point acoustic modes, nonuniform or symmetry-reduced q quadrature, q meshes that change with volume, independent temperature-dependent spectra, anharmonic, electronic or magnetic free energies and anisotropic strain remain outside scope.", "Compiled witnesses establish realizable model examples with positive, negative and zero expansion; they do not certify any material or numerical volume scan.", "The catalog default scheme polyfit_enthalpy_derivative (Phonopy heat_capacity_P_polyfit) evaluates C_V(V_eq)+T*(dV/dT)_p*(dS/dV)_T with polynomial fits of S(V) and of the equilibrium volume. The theorem proves that this expression equals -T*G_TT on the exact model, not that the fits reproduce it. The finite-difference sibling heat_capacity_P_numerical is not verified either. contract_heat_capacity_p_identity has its own binding.", "A producer must reject any imaginary mode at every scanned volume, not only at the selected one. Gamma-point acoustic modes are excluded; a shifted mesh without the Gamma point meets the premises. For polar materials the Born charges and eps_inf of the nonanalytic correction must be recomputed at every volume. The static energy E(V) and the phonon free energy must refer to the same cell at every volume.", "The theorems hold for any T>0 with F_VV>0, including temperatures where the quasiharmonic approximation is physically unreliable, such as strong anharmonicity near melting or soft-mode transitions. They are statements about the model, not about its adequacy." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibQHA", "declarations": [ "OMAI.PhyslibBridge.Response.harmonic_static_qhaHeatCapacityP_eq_omai", "OMAI.PhyslibBridge.Response.harmonic_static_qhaHeatCapacityP_eq_entropy_slope", "OMAI.PhyslibBridge.Response.harmonic_static_qhaHeatCapacityP_gt_iff" ], "source_path": "lean/bridge/OMAIPhyslibQHA.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_heat_capacity_p" }, { "kind": "node", "id": "HeatCapacityConstantP" } ], "source_sha256": "2347e44452e37e3a05131ad636dd20f6ad7ea74dcc60c97fb8a2134f63f68f96" } }, { "id": "harmonic-qha-thermal-expansion", "relation": "conditional", "statement": "For the same windowed Gibbs energy, the expansion (dV/dT)_p/V with V=(dG/dp)_T equals the volumetric expansion of the constant-pressure path, -F_TV/(V*F_VV), and is positive exactly when the heat-capacity-weighted mode Gruneisen mean of the same family is positive.", "targets": [ { "kind": "node", "id": "QHAGibbsEnergy", "uid": "bb10912780eb22d45beff551be14f7f4216f3aae26161e8bfd60cdc051f63f58" }, { "kind": "node", "id": "ThermalExpansion", "uid": "9b42f8aebb6099cdcd910a272a92208881b598688bac7830c3ae0a412152473e" }, { "kind": "operator", "id": "compute_thermal_expansion", "uid": "4e11a6ef1474661ab6ea320a5c99e8774edbe32b5a4dcdb13d97fda4b108995e", "formula_sha256": "7a5492ca3cc69da193307f5f74b36985627e5b4ab5c97bc2851120f491553f93" } ], "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" } ], "conditions": [ "Use the same finite family of full Nat harmonic occupation ensembles as the Cp binding: product counting measure, unweighted oscillator energies, dof=0 and phase-space unit=1, with static energy N_q*E_0(v) attached through an actual singleton ensemble before molar normalization.", "Temperature T and primitive-cell volume v are positive. Every frequency is positive and C2 on an open positive-volume domain. Static energy E_0(v) is C2, arbitrary signed, and independent of temperature. No mode may have zero or imaginary frequency; an unmodified Gamma-centred grid with zero acoustic modes does not satisfy the premises.", "Keep masses, composition, atom and q counts, reduced-q labels, uniform q weights and the Avogadro scale fixed, with one q mesh at every volume. Frequencies depend on volume only.", "The volume family V(T,p) is differentiable at every point of a neighborhood of the evaluation point (T,p), stays in the model domain and satisfies F_V(T,V)+p=0 throughout that neighborhood, with F_VV>0 at the evaluation point. The implicit-function theorem supplies such a family through every stable stationary point; a pointwise stationarity equation alone is insufficient.", "The Gibbs energy is the infimum of F(T,V)+pV over an open volume window (a,b) that contains the selected molar volume and lies in the model domain. The window selects the family when F_VV(T,V)>0 for every V in (a,b) and every T near the evaluation temperature: a scan certifies itself by this convexity of F+pV across the scanned interval. The selected volume must lie strictly inside the scanned range; EOS extrapolation beyond the scanned volumes is not covered. Every sufficiently narrow window around the selected volume also selects the family; a window that reaches another local minimum can select another branch, which is not covered.", "Interpret alpha^qha(G_qha) as the volumetric coefficient (dV/dT)_p/V with V=(dG/dp)_T read off the windowed Gibbs energy. Linear expansion coefficients are one third of this only for cubic isotropic strain and are not this output.", "Embed model values coherently in SI: T in K, p in Pa, E_0 in J per primitive cell, molar F and G in J/mol of primitive cells, v in m^3, V_m in m^3/mol, alpha_V in 1/K, K_T in Pa and capacities in J/(mol K) or J/(m^3 K). Convert Angstrom^3 by 10^-30, serving GPa by 10^9 and eV by 1.602176634e-19 J/eV. These exact-real interpretations do not certify floating-point adapters." ], "conventions": { "model_scope": "Local hydrostatic static-plus-independent-harmonic canonical model with volume-dependent positive frequencies and no independent temperature dependence of the spectrum. The Gibbs energy is a windowed minimum over molar volume of the actual molar Helmholtz surface plus pV. This is an OMAI model-specific extension of actual Physlib ensembles, not an upstream QHA theorem or a numerical QHA solver proof.", "component_scope": "Finite uniform-q grid with all 3*N_atoms branches and fixed reduced-q labels; one scalar volume coordinate along a hydrostatic path, one temperature and one pressure coordinate. Electronic, magnetic, configurational, anharmonic and anisotropic strain contributions are excluded.", "normalization_basis": "Moles of primitive cells with the Avogadro scale N_A=6.02214076e23 mol^-1 for the J/mol embedding. F, G and C_P are molar in that basis and scale with N_A; C_V_vol divides by the primitive cell volume. alpha_V, K_T and Gamma are intensive and do not depend on N_A.", "frequency_convention": "Angular SI frequency with Physlib's CODATA hbar 1.054571817e-34 J s and exact kB in the ensemble energies and heat-capacity weights. Stored linear THz requires omega_SI=2*pi*10^12*nu_THz.", "unit_embedding": "G and F in J/mol, pressure and K_T in Pa, V_m in m^3/mol, alpha_V in 1/K, C_P in J/(mol K) and C_V_vol in J/(m^3 K). Use a common numeric-unit embedding, not raw mixed serving values.", "held_fixed": "Composition, masses, q labels and weights, the q mesh, the Avogadro scale and the static-energy and frequency functions are fixed. C_P and alpha_V differentiate at constant pressure, K_T at constant temperature, C_V at constant volume.", "admissibility": "T>0, v>0, omega>0, N_A>0, N_q>0, local positive isothermal curvature and a window inside the locally convex region. alpha_V and Gamma are sign-free. Totalized values outside the model domain or at T=0 have no asserted physical interpretation.", "reference_energy": "All oscillator zero-point energies remain and shift the selected volume and modulus. The signed temperature-independent static energy enters the actual Hamiltonian and is not silently removed.", "qpoint_weighting": "Uniform full-grid q averaging with the same mesh at every volume; varying primitive volume does not change reduced-q labels or introduce fractional oscillator copies." }, "remaining_obligations": [ "Static-energy, phonon and geometry producers are interpreted inputs, not verified implementations. Numerical volume scans, EOS fits, polynomial or finite-difference derivatives, minimizers and the Phonopy QHA code are not verified; the theorems state what such a producer computes on an exact model, not that a given code does.", "Global minimization over all volumes, first-order transitions between branches, phase coexistence, mechanically unstable branches and the T=0 limit are outside scope.", "Zero or imaginary modes, Gamma-point acoustic modes, nonuniform or symmetry-reduced q quadrature, q meshes that change with volume, independent temperature-dependent spectra, anharmonic, electronic or magnetic free energies and anisotropic strain remain outside scope.", "A producer must reject any imaginary mode at every scanned volume, not only at the selected one. Gamma-point acoustic modes are excluded; a shifted mesh without the Gamma point meets the premises. For polar materials the Born charges and eps_inf of the nonanalytic correction must be recomputed at every volume. The static energy E(V) and the phonon free energy must refer to the same cell at every volume.", "The theorems hold for any T>0 with F_VV>0, including temperatures where the quasiharmonic approximation is physically unreliable, such as strong anharmonicity near melting or soft-mode transitions. They are statements about the model, not about its adequacy.", "Compiled witnesses establish realizable model examples with positive, negative and zero expansion; they do not certify any material or numerical volume scan." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibQHA", "declarations": [ "OMAI.PhyslibBridge.Response.harmonic_static_qhaThermalExpansion_eq_omai", "OMAI.PhyslibBridge.Response.harmonic_static_qhaThermalExpansion_pos_iff", "OMAI.PhyslibBridge.HarmonicVolumeModel.freeEnergy_eq_actual" ], "source_path": "lean/bridge/OMAIPhyslibQHA.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_thermal_expansion" }, { "kind": "node", "id": "ThermalExpansion" } ], "source_sha256": "2347e44452e37e3a05131ad636dd20f6ad7ea74dcc60c97fb8a2134f63f68f96" } }, { "id": "harmonic-qha-thermal-gruneisen-contraction", "relation": "conditional", "statement": "A corollary of the thermal Gruneisen identity of the same model, under the interpretation that the producer's contraction is the C_V-weighted mean of derivative-defined mode Gruneisen parameters at the selected volume, with Physlib canonical oscillator heat capacities as weights. That mean equals alpha_V*K_T/C_V_vol with alpha_V and K_T read off the windowed Gibbs energy. The new content is this interpretation of the opaque producer and the Gibbs-surface reading of alpha_V and K_T; the weighted mean itself is the existing mode-weighted Gamma.", "targets": [ { "kind": "node", "id": "ThermalGruneisen", "uid": "22b1bd519d50d93fd24a919e021bab85aff246464a66a08f8274558834983985" }, { "kind": "operator", "id": "contract_thermal_gruneisen", "uid": "aa554ec7f8a638108f1039bffd5f1c727687f5fd0d018d61d2916e745a5f6f29", "formula_sha256": "01d5dfc469084551b1508829d6a3c4d7f55bccf19207530724fb1f9a56137f26" } ], "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" } ], "conditions": [ "Use the same finite family of full Nat harmonic occupation ensembles as the Cp binding: product counting measure, unweighted oscillator energies, dof=0 and phase-space unit=1, with static energy N_q*E_0(v) attached through an actual singleton ensemble before molar normalization.", "Temperature T and primitive-cell volume v are positive. Every frequency is positive and C2 on an open positive-volume domain. Static energy E_0(v) is C2, arbitrary signed, and independent of temperature. No mode may have zero or imaginary frequency; an unmodified Gamma-centred grid with zero acoustic modes does not satisfy the premises.", "Keep masses, composition, atom and q counts, reduced-q labels, uniform q weights and the Avogadro scale fixed, with one q mesh at every volume. Frequencies depend on volume only.", "The volume family V(T,p) is differentiable at every point of a neighborhood of the evaluation point (T,p), stays in the model domain and satisfies F_V(T,V)+p=0 throughout that neighborhood, with F_VV>0 at the evaluation point. The implicit-function theorem supplies such a family through every stable stationary point; a pointwise stationarity equation alone is insufficient.", "The Gibbs energy is the infimum of F(T,V)+pV over an open volume window (a,b) that contains the selected molar volume and lies in the model domain. The window selects the family when F_VV(T,V)>0 for every V in (a,b) and every T near the evaluation temperature: a scan certifies itself by this convexity of F+pV across the scanned interval. The selected volume must lie strictly inside the scanned range; EOS extrapolation beyond the scanned volumes is not covered. Every sufficiently narrow window around the selected volume also selects the family; a window that reaches another local minimum can select another branch, which is not covered.", "Interpret gamma^qha(gamma_G,omega) as sum_i gamma_i*c_i/sum_i c_i over all modes of the grid, where c_i is the actual canonical heat capacity of an oscillator with frequency omega_i at the evaluation temperature. For the identity with alpha_V*K_T/C_V_vol, gamma_i must be the derivative -v*omega_i_prime(v)/omega_i(v) of the same volume family at the selected cell volume v=V_m/N_A, and C_V_vol=sum(c)/(N_q*v).", "Each labelled branch must be followed continuously through band crossings; sorted band order is not sufficient for the derivative-defined mode parameters.", "Embed model values coherently in SI: T in K, p in Pa, E_0 in J per primitive cell, molar F and G in J/mol of primitive cells, v in m^3, V_m in m^3/mol, alpha_V in 1/K, K_T in Pa and capacities in J/(mol K) or J/(m^3 K). Convert Angstrom^3 by 10^-30, serving GPa by 10^9 and eV by 1.602176634e-19 J/eV. These exact-real interpretations do not certify floating-point adapters." ], "conventions": { "model_scope": "Local hydrostatic static-plus-independent-harmonic canonical model with volume-dependent positive frequencies and no independent temperature dependence of the spectrum. The Gibbs energy is a windowed minimum over molar volume of the actual molar Helmholtz surface plus pV. This is an OMAI model-specific extension of actual Physlib ensembles, not an upstream QHA theorem or a numerical QHA solver proof.", "component_scope": "Finite uniform-q grid with all 3*N_atoms branches and fixed reduced-q labels; one scalar volume coordinate along a hydrostatic path, one temperature and one pressure coordinate. Electronic, magnetic, configurational, anharmonic and anisotropic strain contributions are excluded.", "normalization_basis": "Moles of primitive cells with the Avogadro scale N_A=6.02214076e23 mol^-1 for the J/mol embedding. F, G and C_P are molar in that basis and scale with N_A; C_V_vol divides by the primitive cell volume. alpha_V, K_T and Gamma are intensive and do not depend on N_A.", "frequency_convention": "Angular SI frequency with Physlib's CODATA hbar 1.054571817e-34 J s and exact kB in the ensemble energies and heat-capacity weights. Stored linear THz requires omega_SI=2*pi*10^12*nu_THz.", "unit_embedding": "G and F in J/mol, pressure and K_T in Pa, V_m in m^3/mol, alpha_V in 1/K, C_P in J/(mol K) and C_V_vol in J/(m^3 K). Use a common numeric-unit embedding, not raw mixed serving values.", "held_fixed": "Composition, masses, q labels and weights, the q mesh, the Avogadro scale and the static-energy and frequency functions are fixed. C_P and alpha_V differentiate at constant pressure, K_T at constant temperature, C_V at constant volume.", "admissibility": "T>0, v>0, omega>0, N_A>0, N_q>0, local positive isothermal curvature and a window inside the locally convex region. alpha_V and Gamma are sign-free. Totalized values outside the model domain or at T=0 have no asserted physical interpretation.", "reference_energy": "All oscillator zero-point energies remain and shift the selected volume and modulus. The signed temperature-independent static energy enters the actual Hamiltonian and is not silently removed.", "qpoint_weighting": "Uniform full-grid q averaging with the same mesh at every volume; varying primitive volume does not change reduced-q labels or introduce fractional oscillator copies." }, "remaining_obligations": [ "Static-energy, phonon and geometry producers are interpreted inputs, not verified implementations. Numerical volume scans, EOS fits, polynomial or finite-difference derivatives, minimizers and the Phonopy QHA code are not verified; the theorems state what such a producer computes on an exact model, not that a given code does.", "Global minimization over all volumes, first-order transitions between branches, phase coexistence, mechanically unstable branches and the T=0 limit are outside scope.", "Zero or imaginary modes, Gamma-point acoustic modes, nonuniform or symmetry-reduced q quadrature, q meshes that change with volume, independent temperature-dependent spectra, anharmonic, electronic or magnetic free energies and anisotropic strain remain outside scope.", "Compiled witnesses establish realizable model examples with positive, negative and zero expansion; they do not certify any material or numerical volume scan.", "FC3 (Maradudin-Fein) and finite-difference mode Gruneisen producers, including compute_gruneisen, are not shown to equal the derivative-defined gamma_i. Phonopy gruneisen_temperature is a fitted quotient and does not consume mode parameters. FC3 or finite-difference gammas evaluated at the reference geometry differ from gammas at the selected volume.", "A producer must reject any imaginary mode at every scanned volume, not only at the selected one. Gamma-point acoustic modes are excluded; a shifted mesh without the Gamma point meets the premises. For polar materials the Born charges and eps_inf of the nonanalytic correction must be recomputed at every volume. The static energy E(V) and the phonon free energy must refer to the same cell at every volume.", "The theorems hold for any T>0 with F_VV>0, including temperatures where the quasiharmonic approximation is physically unreliable, such as strong anharmonicity near melting or soft-mode transitions. They are statements about the model, not about its adequacy." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibQHA", "declarations": [ "OMAI.PhyslibBridge.Response.harmonic_static_qhaThermalGruneisen_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibQHA.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_thermal_gruneisen" }, { "kind": "node", "id": "ThermalGruneisen" } ], "source_sha256": "2347e44452e37e3a05131ad636dd20f6ad7ea74dcc60c97fb8a2134f63f68f96" } }, { "id": "harmonic-static-cp-identity", "relation": "conditional", "statement": "For the static-plus-harmonic Physlib ensemble, the temperature derivative of actual molar enthalpy along a local stable constant-pressure branch equals the OMAI Cp identity. Cv, volume, expansion and isothermal bulk modulus must belong to the same molar free-energy surface.", "targets": [ { "kind": "node", "id": "BulkModulus", "uid": "c0a13e64c02811f49b951271c894373267937676e0c657fc9ef09fac72558ced" }, { "kind": "node", "id": "HeatCapacityConstantP", "uid": "08057e4554a3420c933329b3821de9295e2b5950191e23cafe310b0a15adc7b6" }, { "kind": "node", "id": "MolarHeatCapacity", "uid": "4fa2d8e16bde42f794f150f5d149e5d645d63f7ddaac59cef79bd41771b13c17" }, { "kind": "node", "id": "MolarVolume", "uid": "e4ae49aad73336225c26fd3cf5d8dc2f4477b02b7d8bad22c9c74fd40a2482b2" }, { "kind": "node", "id": "Temperature", "uid": "63f9bd32d94d77e12afbb96f5b1b7a870de6b1f5805213b7f3c3ff227da301f0" }, { "kind": "node", "id": "ThermalExpansion", "uid": "9b42f8aebb6099cdcd910a272a92208881b598688bac7830c3ae0a412152473e" }, { "kind": "operator", "id": "contract_heat_capacity_p_identity", "uid": "79a11e58595c8b262c104330dcfc622f13a787da6bae856cf4f40b57a6a93639", "formula_sha256": "97ed2e0d3a11efad01a3ab0017d21be3da3009446b0b9364673fad4b8fdcb0f1" } ], "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.meanEnergy", "kind": "def", "line": 519, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L519" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.thermodynamicEntropy", "kind": "def", "line": 800, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L800" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.meanEnergyT", "kind": "def", "line": 409, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L409" } ], "conditions": [ "Use a finite heterogeneous family of full Nat harmonic occupation ensembles, product counting measure, unweighted oscillator energies, dof=0 and phase-space unit=1. Attach static energy N_q*E_0(v) through an actual singleton ensemble before molar normalization.", "Temperature T and primitive-cell volume v are positive. Every frequency is positive and C2 on an open positive-volume domain. Static energy E_0(v) is C2, arbitrary signed, and independent of temperature.", "Keep masses, composition, atom and q counts, reduced-q labels, uniform q weights and Avogadro scale fixed. Frequencies depend on volume but have no independent temperature dependence.", "Use molar volume V_m=N_A*v. Let F_static_joint be the actual free energy after attaching the N_q*E_0(v) singleton, and F_harmonic_joint the unshifted harmonic free energy. Then F_mol=(N_A/N_q)*F_static_joint=N_A*E_0(v)+(N_A/N_q)*F_harmonic_joint. Evaluate all quantities at the same temperature and equilibrium volume, per mole of the same primitive cells, not per mole of atoms.", "The supplied volume path is differentiable at the evaluation temperature and satisfies domain membership and F_V+p_ext=0 throughout a neighborhood, with constant external pressure. Require F_VV>0 at the evaluation point; the proved C2 regularity permits restriction to a stable neighborhood. A pointwise stationarity equation alone is insufficient.", "Interpret expansion as V_m_prime/V_m and bulk modulus as the isothermal derivative V_m*F_VV. Cp is the derivative of actual molar internal energy plus p_ext*V_m along that path; Cv is the fixed-volume capacity of the same actual ensemble.", "Embed model values coherently in SI: T in K, E_0 in J per primitive cell, molar F/U in J/mol of those cells, v in m^3, V_m in m^3/mol, alpha in 1/K, pressure and isothermal bulk modulus in Pa, and capacities in J/(mol K). Convert cell Angstrom^3 by 10^-30 to m^3, canonical pressure eV/Angstrom^3 by 1.602176634e11 to Pa, and serving GPa by 10^9 to Pa. Static energies served in eV require 1.602176634e-19 J/eV. These exact-real interpretations do not certify floating-point adapters." ], "conventions": { "model_scope": "Local hydrostatic static-plus-independent-harmonic canonical model with volume-dependent positive frequencies. This is an OMAI model-specific extension of actual Physlib ensembles, not an existing upstream Cp theorem or a numerical QHA solver proof.", "component_scope": "Finite uniform-q grid with all 3*N_atoms branches, fixed reduced-q labels, scalar volumetric expansion and isothermal bulk response.", "normalization_basis": "Moles of primitive cells. The Hamiltonian static offset is N_q*E_0(v); N_A/N_q is applied to evaluated thermodynamic quantities, giving N_A*E_0(v) in molar F and U. There is no branch-count divisor.", "frequency_convention": "Angular SI frequency with exact upstream hbar and kB. Stored linear THz requires omega_SI=2*pi*10^12*nu_THz; finite-precision conversion code is not certified.", "unit_embedding": "F and U in J/mol, capacities J/(mol K), primitive-cell v in m^3 and molar V_m in m^3/mol; pressure and isothermal bulk modulus in Pa. Use a common numeric-unit embedding, not raw mixed serving values.", "held_fixed": "Composition, masses, q labels/weights, Avogadro scale and the static-energy/frequency functions are fixed. Cv differentiates at fixed volume; Cp differentiates along the locally constant-pressure path.", "admissibility": "T>0, v>0, omega>0, N_A>0, N_q>0 and local positive isothermal curvature. Totalized values outside the model domain have no asserted physical interpretation.", "reference_energy": "All oscillator zero-point energies remain. The signed temperature-independent static energy enters the actual Hamiltonian and is not silently removed.", "qpoint_weighting": "Uniform full-grid q averaging; varying primitive volume does not change reduced-q labels or introduce fractional oscillator copies." }, "remaining_obligations": [ "Static-energy, phonon, geometry, thermal-expansion and bulk-modulus producers are interpreted inputs, not verified implementations. The theorem does not validate EOS fits, minimizers, finite-difference/polyfit derivatives or the sibling compute_heat_capacity_p producer.", "The equilibrium branch and common-potential input interpretation remain explicit local hypotheses of the general theorem. The compiled nonzero-expansion and zero-response witnesses establish realizable model examples; they do not certify any particular material or numerical volume scan.", "Zero or imaginary modes, independent temperature-dependent spectra, nonuniform q quadrature, intermode coupling, phase coexistence and mechanically unstable branches remain outside scope.", "No claim is made about FC3 mode gammas or the opaque thermal-Gruneisen producer. A fitted thermodynamic quotient is not independent evidence for mode-weighted gamma agreement." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicResponse", "declarations": [ "OMAI.PhyslibBridge.Response.harmonic_static_heatCapacityP_eq_omai", "OMAI.PhyslibBridge.Response.harmonic_static_heatCapacityP_ge_heatCapacityV" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicResponse.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_heat_capacity_p_identity" }, { "kind": "node", "id": "HeatCapacityConstantP" } ], "source_sha256": "1914f2a738ae2ffaa4b35cfce04dc3a880f2d67cbf89a4ff47c7f15ec8a5f9b0" } }, { "id": "harmonic-static-thermal-gruneisen-identity", "relation": "conditional", "statement": "For the static-plus-harmonic Physlib ensemble on a local stable constant-pressure branch, the heat-capacity-weighted mean of mode Gruneisen parameters, defined from the volume derivatives of the mode frequencies and weighted by the actual canonical mode heat capacities, equals the OMAI identity alpha_V*K/C_V_vol. Expansion, isothermal bulk modulus and volumetric heat capacity must belong to the same free-energy surface, temperature and equilibrium volume.", "targets": [ { "kind": "node", "id": "BulkModulus", "uid": "c0a13e64c02811f49b951271c894373267937676e0c657fc9ef09fac72558ced" }, { "kind": "node", "id": "ThermalExpansion", "uid": "9b42f8aebb6099cdcd910a272a92208881b598688bac7830c3ae0a412152473e" }, { "kind": "node", "id": "ThermalGruneisen", "uid": "22b1bd519d50d93fd24a919e021bab85aff246464a66a08f8274558834983985" }, { "kind": "node", "id": "VolumetricHeatCapacity", "uid": "29e324b97f31cdb7d3cd8ef6aec5668fff111fd7ee96bd19dd0426b8728c1d47" }, { "kind": "operator", "id": "contract_thermal_gruneisen_identity", "uid": "1348e12defce337587f9c2b152a3a1c88f9d27ec8cc48fc30167aebadd907043", "formula_sha256": "d84872a220cba0af3fed426df7df8137e40c15bf547cdb3a4132d5e20af18f23" } ], "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.helmholtzFreeEnergy", "kind": "def", "line": 680, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L680" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Basic", "name": "CanonicalEnsemble.meanEnergy", "kind": "def", "line": 519, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean#L519" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.meanEnergyT", "kind": "def", "line": 409, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L409" }, { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" } ], "conditions": [ "Use the same finite family of full Nat harmonic occupation ensembles as the Cp binding: product counting measure, unweighted oscillator energies, dof=0 and phase-space unit=1, with static energy N_q*E_0(v) attached through an actual singleton ensemble before molar normalization.", "Temperature T and primitive-cell volume v are positive. Every frequency is positive and C2 on an open positive-volume domain. Static energy E_0(v) is C2, arbitrary signed, and independent of temperature. No mode may have zero or imaginary frequency; an unmodified Gamma-centred grid with zero acoustic modes does not satisfy the premises.", "Keep masses, composition, atom and q counts, reduced-q labels, uniform q weights and the Avogadro scale fixed, with one q mesh at every volume. Frequencies depend on volume only. Each labelled branch must be followed continuously through band crossings; sorted band order is not sufficient.", "Define the mode Gruneisen parameter as gamma_i(v)=-v*omega_i_prime(v)/omega_i(v) from the volume family itself. Define Gamma(T,v)=sum_i gamma_i*c_i/sum_i c_i, where c_i is the actual derivative-defined canonical heat capacity of mode i at fixed v. Gamma does not use expansion, bulk modulus, the branch, the free energy or the quotient alpha_V*K/C_V_vol.", "Evaluate Gamma, expansion, bulk modulus and volumetric heat capacity at the same temperature and at the branch volume v_B=V_m/N_A, per the same primitive cell that carries E_0(v) and the phonon grid. Supplied static energies, volumes and phonons must refer to that one cell.", "The supplied volume path is differentiable at the evaluation temperature and satisfies domain membership and F_V+p_ext=0 throughout a neighborhood, with constant external pressure. Require F_VV>0 at the evaluation point. A pointwise stationarity equation alone is insufficient.", "Interpret ThermalExpansion as the volumetric coefficient V_m_prime/V_m along that path, BulkModulus as the isothermal modulus V_m*F_VV at the same temperature, and VolumetricHeatCapacity as sum(c)/(N_q*v) at fixed v, equal to C_V_mol/V_m. Linear expansion, adiabatic or elastic moduli, static-only curvature and constant-pressure capacities are not these inputs.", "Use the volumetric form only. The molar form alpha_V*K*V_m/C_V_mol is equivalent only with V_m and C_V_mol per mole of the same primitive cells. An adiabatic modulus paired with C_V_vol gives Gamma*(1+alpha_V*Gamma*T), not Gamma.", "Supporting result, not a separate subject: at every point of the model domain, without a branch, the temperature derivative of pressure at fixed volume equals Gamma*C_V_vol.", "Gamma, every gamma_i and alpha_V may be negative or zero. No statement is made at T=0, where Gamma is a limit of a ratio of vanishing capacities.", "Embed model values coherently in SI: T in K, E_0 in J per primitive cell, v in m^3, V_m in m^3/mol, alpha_V in 1/K, isothermal bulk modulus in Pa and C_V_vol in J/(m^3 K). Convert Angstrom^3 by 10^-30, serving GPa by 10^9 and eV by 1.602176634e-19 J/eV. These exact-real interpretations do not certify floating-point adapters." ], "conventions": { "model_scope": "Local hydrostatic static-plus-independent-harmonic canonical model with volume-dependent positive frequencies and no independent temperature dependence of the spectrum. Gamma is the heat-capacity-weighted mean of derivative-defined mode Gruneisen parameters. This is an OMAI model-specific extension of actual Physlib ensembles, not an upstream Gruneisen theorem or a numerical QHA solver proof.", "component_scope": "Finite uniform-q grid with all 3*N_atoms branches, continuously labelled branches and fixed reduced-q labels; one scalar volume coordinate along a hydrostatic path. Electronic, magnetic, configurational and anisotropic strain contributions are excluded.", "normalization_basis": "One primitive cell carries E_0(v), the volume coordinate and the phonon grid. Molar means per mole of those primitive cells and volumetric means per primitive-cell volume. Gamma, alpha_V, K and C_V_vol are intensive: the Avogadro scale and N_q cancel, and the identity uses the volumetric form only.", "frequency_convention": "Angular SI frequency with exact upstream hbar and kB in the heat-capacity weights. Stored linear THz requires omega_SI=2*pi*10^12*nu_THz. Mode Gruneisen parameters are logarithmic derivatives and do not depend on a constant frequency scale; the weights do.", "unit_embedding": "Coherent SI: alpha_V in 1/K, isothermal bulk modulus in Pa, C_V_vol in J/(m^3 K), primitive-cell v in m^3 and V_m in m^3/mol; Gamma and gamma_i are dimensionless. Use a common numeric-unit embedding, not raw mixed serving values.", "held_fixed": "Composition, masses, q labels and weights, the q mesh, the Avogadro scale and the static-energy and frequency functions are fixed. gamma_i and C_V_vol are taken at fixed v; the bulk modulus at fixed temperature; the expansion along the locally constant-pressure path.", "admissibility": "T>0, v>0, omega>0, N_A>0, N_q>0 and local positive isothermal curvature on the branch. Gamma, gamma_i and alpha_V are sign-free. Totalized values outside the model domain or at T=0 have no asserted physical interpretation.", "reference_energy": "All oscillator zero-point energies remain and shift the equilibrium volume and bulk modulus. The signed temperature-independent static energy enters the actual Hamiltonian and is not silently removed; neither contributes to the mixed temperature-volume derivative.", "qpoint_weighting": "Uniform full-grid q averaging with the same mesh at every volume; varying primitive volume does not change reduced-q labels or introduce fractional oscillator copies. Symmetry-reduced, tetrahedron or other nonuniform weights are outside this definition." }, "remaining_obligations": [ "Static-energy, phonon, geometry, thermal-expansion, bulk-modulus and heat-capacity producers are interpreted inputs, not verified implementations. The theorem does not validate EOS fits, minimizers, finite-difference or polyfit derivatives, or the numerical QHA producers compute_thermal_expansion, compute_bulk_modulus_qha, compute_qha_gibbs and compute_heat_capacity_p.", "No claim is made about FC3 (Maradudin-Fein) or finite-difference mode Gruneisen producers, including compute_gruneisen. They estimate frequency volume derivatives at a reference volume and are not shown to equal the derivative-defined gamma_i used here.", "Phonopy PhonopyQHA.gruneisen_temperature evaluates a fitted alpha_V*B_T*V/C_V quotient with a finite-difference expansion, a polynomial C_V fit and zero fallbacks. It does not consume mode Gruneisen parameters and is not independent evidence for agreement with the mode-weighted Gamma.", "The opaque weighted contraction contract_thermal_gruneisen is not covered; it needs its own reviewed interpretation.", "The equilibrium branch and the common-potential input interpretation remain explicit local hypotheses. Compiled witnesses establish realizable model examples, including distinct mode Gruneisen parameters and a constant-frequency zero-response control with a single q point; they do not certify any material or numerical volume scan.", "Zero or imaginary modes, Gamma-point acoustic modes, nonuniform or symmetry-reduced q quadrature, q meshes that change with volume, the thermodynamic limit, independent temperature-dependent spectra, intermode coupling, anisotropic strain and tensor Gruneisen parameters, electronic or magnetic free energies, phase coexistence and mechanically unstable branches remain outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibThermalGruneisen", "declarations": [ "OMAI.PhyslibBridge.Response.harmonic_static_thermalGruneisen_eq_omai" ], "source_path": "lean/bridge/OMAIPhyslibThermalGruneisen.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_thermal_gruneisen_identity" }, { "kind": "node", "id": "ThermalGruneisen" } ], "source_sha256": "6669b924a8682f135cee4daeda9d2d76b5418aa1ee30117dd7975dc86dddece9" } }, { "id": "harmonic-volumetric-heat-capacity", "targets": [ { "kind": "node", "id": "VolumetricHeatCapacity", "uid": "29e324b97f31cdb7d3cd8ef6aec5668fff111fd7ee96bd19dd0426b8728c1d47" }, { "kind": "operator", "id": "contract_volumetric_heat_capacity", "uid": "239c7dd3370e7b7e4456f1ade35c5d81edba69efda786d25e8e1e74f6fa297d9", "formula_sha256": "e89d3db8b665f634538055816c2e1336259f88bcd007badcf318c802d2b3ff92" } ], "relation": "conditional", "statement": "The sum of actual Physlib single-mode heat capacities divided by N_q*V_cell, with positive primitive-cell volume V_cell, equals the OMAI volumetric heat-capacity formula. The same grid satisfies C_vol=C_mol/(N_A*V_cell).", "sources": [ { "module": "Physlib.StatisticalMechanics.CanonicalEnsemble.Lemmas", "name": "CanonicalEnsemble.heatCapacity", "kind": "def", "line": 417, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean#L417" }, { "module": "Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.TISE", "name": "QuantumMechanics.OneDimension.HarmonicOscillator.eigenValue", "kind": "def", "line": 29, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/TISE.lean#L29" } ], "conditions": [ "Use a finite grid Fin N_q x Fin (3*N_atoms), with positive natural N_q and N_atoms. Fin labels map to OMAI 1-based labels by i.val+1.", "Each grid entry is an upstream positive-frequency harmonic oscillator with exact upstream hbar and kB; all entries share one positive absolute temperature.", "All modes must have strictly positive frequencies. An unmodified Gamma grid with zero acoustic modes does not satisfy the premises.", "Hold the mode spectrum, weights and primitive-cell basis fixed; convert stored linear THz to angular frequency.", "V_cell is the positive primitive-cell volume in m^3, not a supercell volume; convert stored Angstrom^3 by the exact factor 10^-30.", "Use sum(c)/(N_q*V_cell), with no Avogadro factor or branch-count divisor. The molar comparison uses the same positive N_A and primitive-cell basis." ], "conventions": { "model_scope": "Finite uniform-q contraction of actual derivative-defined Physlib single-mode heat capacities.", "component_scope": "N_q q points and exactly 3*N_atoms branches per primitive-cell basis.", "normalization_basis": "Sum all 3*N_atoms branches; divide by N_q times primitive-cell volume, not by atom or branch count.", "frequency_convention": "Symbolic omega is angular rad/s. Canonical linear THz requires omega=2*pi*10^12*nu_THz, not value identity.", "unit_embedding": "Per-mode heat capacity J/K and V_cell in m^3 give J/(m^3 K); Angstrom^3 converts exactly by 10^-30. Molar comparison uses J/(mol K) per mole of primitive cells.", "held_fixed": "Mode spectrum, q weights, and cell volume fixed when differentiating in T.", "admissibility": "T>0, omega>0, hbar>0, kB>0. Zero, imaginary and unstable modes are outside the physical model; totalized Lean expressions do not give them a physical interpretation.", "reference_energy": "Each spectrum includes zero-point energy; its temperature derivative contributes no additive heat-capacity term.", "qpoint_weighting": "Uniform q weight 1/N_q, N_q>0; sum all 3*N_atoms branches." }, "remaining_obligations": [ "Primitive-cell geometry and phonon frequencies remain interpreted inputs, not verified producers. Nonuniform q, zero/imaginary modes, thermodynamic limits and finite-precision implementations remain outside scope." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibHarmonicDensity", "declarations": [ "OMAI.PhyslibBridge.harmonic_volumetric_heatCapacity_eq_omai", "OMAI.PhyslibBridge.harmonicVolumetricHeatCapacity_eq_molar_div", "OMAI.PhyslibBridge.uniformVolumetricContraction_ofAngstromCubed", "OMAI.PhyslibBridge.harmonicHeatCapacity_angularSI_eq_linearTHz" ], "source_path": "lean/bridge/OMAIPhyslibHarmonicDensity.lean", "source_sha256": "cf4d59af5b660a1fc73b079d63451f269081e1bd9a3018047e1f6f41517013ea", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_volumetric_heat_capacity" }, { "kind": "node", "id": "VolumetricHeatCapacity" } ] } }, { "id": "hellmann-feynman-force-balance", "relation": "local", "targets": [ { "kind": "node", "id": "Forces", "uid": "9c5a2fdedab6b668ccf5f2043fb4c29a982bcde5063fd6ac8633ee2c7858ecbf" }, { "kind": "operator", "id": "compute_forces_hf", "uid": "59ae54527b2e735e5408e66eadb81c9f1fa29720ea39207564e388a44cd1b04c", "formula_sha256": "198daf948e48b420f367655f4ab063d3da5f54dd475468ce249711afdef38d0b" } ], "statement": "For a total energy differentiable at the configuration and invariant under rigid translations of all atoms, the catalog forces F_{i,alpha} = -dE/dR_{i,alpha} sum to zero over the atoms in every Cartesian direction. For an energy invariant under rotations in the (alpha, beta) plane, the total torque in that plane vanishes: sum_i (R_{i alpha} F_{i beta} - R_{i beta} F_{i alpha}) = 0. A compiled two-atom energy |R_1 - R_0|^2 satisfies both invariances with a nonzero force, so neither statement is vacuous.", "sources": [], "conditions": [ "E is the total energy as a function of all atomic Cartesian coordinates of a finite set of atoms, and the force is its exact partial derivative, the Pulay- and nonlocal-corrected total that codes report, not the bare Hellmann-Feynman term.", "Translation invariance holds for an isolated system and for a periodic cell, where every atom and every image shifts by the same vector and the lattice vectors are unchanged. A real-space grid that breaks it (the egg-box effect) gives a nonzero force sum; the theorem then says the residual measures that breaking.", "Rotation invariance about the origin holds for an isolated molecule or cluster. It does not hold for atoms rotated inside a fixed periodic lattice, so the torque result does not apply to periodic cells.", "The energy is differentiable at the configuration, so the forces are its actual partial derivatives; without that hypothesis Mathlib's derivative is zero and the statements would hold only through that zero value." ], "conventions": { "model_scope": "Exact derivatives of one energy surface of a finite set of atoms; a local theorem over Mathlib with no Physlib model.", "component_scope": "All atoms and all three Cartesian components; the rotation statement is per coordinate plane.", "normalization_basis": "Forces per atom; the energy is the total energy of the configuration, not per atom.", "frequency_convention": "Not applicable; no frequencies enter these derivatives.", "unit_embedding": "Coherent SI: energy in J, positions in m, forces in N; eV/A forces convert with 1 eV/A = 1.602176634e-9 N.", "held_fixed": "The lattice vectors of a periodic cell are unchanged by a rigid translation of the atoms; nothing else is held fixed.", "admissibility": "The energy differentiable at the configuration; translation invariance at the configuration for the force sum; invariance under rotations in the stated plane for the torque.", "reference_energy": "An additive constant in E changes no force.", "qpoint_weighting": "Not applicable; no Brillouin-zone sums enter these derivatives." }, "remaining_obligations": [ "SCF convergence, smearing (forces are then derivatives of the free energy), basis incompleteness and Pulay corrections, and real-space-grid egg-box errors in DFT producers are not verified.", "No statement is made about the accuracy of the energy surface itself or about forces from machine-learned potentials." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIAtomisticDerivatives", "declarations": [ "OMAI.PhyslibBridge.Atomistic.hfForce_sum_eq_zero", "OMAI.PhyslibBridge.Atomistic.hfForce_torque_eq_zero", "OMAI.PhyslibBridge.Atomistic.pairEnergy_force", "OMAI.PhyslibBridge.Atomistic.pairEnergy_translate", "OMAI.PhyslibBridge.Atomistic.pairEnergy_rotatePlane" ], "source_path": "lean/bridge/OMAIAtomisticDerivatives.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_forces_hf" }, { "kind": "node", "id": "Forces" } ], "source_sha256": "c4b5049d77057a88a806c759d04159c752f5aeb8e30f2eb6ac10908c5a0e41a9" } }, { "id": "hyperelastic-elastic-constants", "relation": "local", "targets": [ { "kind": "node", "id": "ElasticConstants", "uid": "01043ab212fd87e4ba9493caf709ab9a7ccc86ec7a0047a0512af12eca1c391e" }, { "kind": "operator", "id": "compute_elastic_constants", "uid": "3e995930ac921247acde0b3724b91c698f5c8e93c991635155d50a9c8039fe85", "formula_sha256": "266bfe9ac04cee4e494fb4002abaffafab63319c53a23218ce1196c566832d6c" } ], "statement": "For a cell energy twice continuously differentiable at a stress-free reference cell and invariant under rigid rotations of the cells near the reference, the catalog elastic constants C_abcd = -d sigma_ab / d eps_cd have the major symmetry C_abcd = C_cdab and both minor symmetries C_abcd = C_bacd = C_abdc. If they also store positive energy for every nonzero symmetric strain, the catalog's Voigt bulk and shear moduli are positive and its Poisson ratio lies strictly between -1 and 1/2. The St Venant-Kirchhoff energy V0 (lam/2 (tr E)^2 + mu tr(E E)) of the Green-Lagrange strain E = (F^T F - 1)/2 is a compiled witness: rotation invariant, a polynomial in F (compiled as twice continuously differentiable) and stress-free at the reference, with catalog elastic constants (V0/V)(lam delta_ab delta_cd + mu (delta_ac delta_bd + delta_ad delta_bc)) at the reference, positive definite for lam >= 0, mu > 0 and V = V0 > 0, so both theorems apply to it.", "sources": [], "conditions": [ "The elastic constants derive from the compressive-positive catalog stress of one cell energy E(F), F = 1 + eps, so C = (1/V) d^2E / d eps d eps at the reference cell.", "The minor symmetries need a stress-free reference. At a pre-stressed reference the minor symmetries fail by residual-stress terms, and the stress-strain coefficients a code fits there lose the major symmetry as well; only the energy second derivative keeps it.", "For relaxed-ion constants E(F) must already be minimized over the internal coordinates at each strain.", "Rotation invariance is needed only on a neighbourhood of the reference cell (objectivity near F = 1), not at singular deformation gradients; the witness is invariant everywhere." ], "conventions": { "model_scope": "Second derivatives of one rotation-invariant cell energy at a stress-free reference; a local theorem over Mathlib with no Physlib model.", "component_scope": "Full fourth-rank tensor on Cartesian indices; Voigt 6x6 storage must be expanded with the minor symmetries.", "normalization_basis": "Per unit reference volume.", "frequency_convention": "Not applicable; no frequencies enter these derivatives.", "unit_embedding": "Coherent SI: elastic constants in Pa; 1 GPa = 1e9 Pa.", "held_fixed": "Temperature is not involved; internal coordinates as prescribed by the cell energy.", "admissibility": "E twice continuously differentiable at the reference, invariant under rotations of the cells near it, with zero stress there; positive strain energy for the Voigt bounds; lam >= 0, mu > 0 and V = V0 > 0 for the witness bounds.", "reference_energy": "An additive constant in E changes no elastic constant.", "qpoint_weighting": "Not applicable; no Brillouin-zone sums enter these derivatives." }, "remaining_obligations": [ "Finite-strain fits of stress differences (LAMMPS ELASTIC, DFT stress-strain) estimate the derivative and are not verified, nor are the strain magnitudes and symmetrization they use.", "The compiled St Venant-Kirchhoff witness is isotropic; no anisotropic hyperelastic energy is compiled, and the witness says nothing about any code's fitted constants.", "A relaxed DFT cell is stress-free only to its pressure threshold (QE press_conv_thr default 0.5 kbar) plus the Pulay stress of a fixed cutoff, and a cell at the experimental lattice constant carries residual stress of order GPa; the minor-symmetry violation is then of order the residual stress over C." ], "evidence": { "kind": "compiled_bridge", "module": "OMAICellElasticity", "declarations": [ "OMAI.PhyslibBridge.CellElasticity.elasticConstants_symmetries", "OMAI.PhyslibBridge.CellElasticity.voigt_pos_of_stable_reference", "OMAI.PhyslibBridge.CellElasticity.svkEnergy_elasticConstants_symmetries", "OMAI.PhyslibBridge.CellElasticity.svkEnergy_elasticConstants", "OMAI.PhyslibBridge.CellElasticity.svkEnergy_voigt_pos" ], "source_path": "lean/bridge/OMAICellElasticity.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_elastic_constants" }, { "kind": "node", "id": "ElasticConstants" } ], "source_sha256": "c0663d570244e4c53317c1c7cbb193d5acd2429766316cc79685474bc7989e3d" } }, { "id": "linear-elasticity-voigt-moduli", "relation": "local", "targets": [ { "kind": "node", "id": "BulkModulus", "uid": "c0a13e64c02811f49b951271c894373267937676e0c657fc9ef09fac72558ced" }, { "kind": "node", "id": "PoissonRatio", "uid": "e95ba3f5810f8554656c148dae26520cf3258cace0b951dd75e69ee8c552c132" }, { "kind": "node", "id": "ShearModulus", "uid": "1ab85488c90143e38b94e27f93ab96571f381e7b09d507fa7a38216ffd34c5ec" }, { "kind": "node", "id": "YoungsModulus", "uid": "022d18213f7d63713ef4fddb484e1ae6d898873ce20377176013f620225637d8" }, { "kind": "operator", "id": "compute_bulk_modulus_eos", "uid": "e78d5769e8a4718966dee70a82eb5ac9622b35e20edf471ee1d8dc88ab40868c", "formula_sha256": "486f990eb13996a627d4274aada76d2ba5b3cdff7518587837b45ee121e12ace" }, { "kind": "operator", "id": "contract_bulk_modulus", "uid": "646fa795ffeed1fc8935994dd17159062c596fc0952ebb2a3159b3a1a74946fa", "formula_sha256": "461564a49eea3bd54eae3039dc5c24685f31ed2fbd89c5306cafadf7c74d2122" }, { "kind": "operator", "id": "contract_poisson_ratio", "uid": "1374919bed5b2ab25f700faf12ff8dc32b6fa3296fe92dd12092e3b9edd39e54", "formula_sha256": "b9c1551011306c38dd5511cafba2afc0051f45b276aa1f7b4e5d8d5a0228a49b" }, { "kind": "operator", "id": "contract_shear_modulus", "uid": "3466399c40c467b80ca4b9b3a83c9c5b0d73df41ed0295e3a4d6685683972909", "formula_sha256": "a6d3c29f0a742deb9c3acf0be98694e08dd2ee4ce9e40e86b4bfb7d0d4931d69" }, { "kind": "operator", "id": "contract_youngs_modulus", "uid": "1b6ab3dbb0648151cd191a9a98d25d103e1a8461958a2f04c14dfa4627c248cf", "formula_sha256": "375c6fffee472f619a478562c1a12bc1fd48b7dc55a1a13abcffbaed8dfcc9e6" } ], "statement": "For every stiffness tensor with the first minor symmetry that stores positive energy for every nonzero symmetric strain, the catalog's Voigt bulk and shear moduli are positive and its Poisson ratio lies strictly between -1 and 1/2. For an isotropic tensor, positivity of the Voigt bulk and shear moduli is equivalent to stability. For a cubic tensor with C12 <= C11 and C44 >= 0 the curvature V0 E''(V0) of the cell energy minimized over cell shape at each volume is K. Under uniform scaling of the cell without shape relaxation the curvature is the Voigt K for every tensor.", "sources": [], "conditions": [ "ElasticConstants is the full fourth-rank stiffness tensor C_abcd on Cartesian indices in one orientation. Voigt 6x6 storage is expanded with the minor symmetries before the catalog sums apply; Mandel or Kelvin storage, with its factors sqrt(2) and 2 on the shear block, must be converted first.", "The Lean stress sigma = C : e is tensile-positive, the negative of the catalog's compressive-positive Stress node, consistent with compute_elastic_constants defining C = -d sigma_OMAI / d eps.", "The stability results require the first minor symmetry C_abcd = C_bacd and positive strain energy for every nonzero symmetric strain. The isotropic results use C_abcd = lam d_ab d_cd + mu (d_ac d_bd + d_ad d_bc); the cubic results use Voigt constants C11, C12, C44 with the cube axes along the Cartesian axes.", "The catalog's Poisson ratio and Young's modulus take BulkModulus then ShearModulus. For an anisotropic crystal, E and nu computed from its Voigt K and G are Voigt-polycrystal estimates, neither Voigt-Reuss-Hill averages nor single-crystal directional values.", "The equation-of-state results evaluate compute_bulk_modulus_eos at the reference volume V0 of a linear-elastic cell with energy E0 + V0 * W(e) / 2, W(e) = sum_abcd C_abcd e_ab e_cd, and volumetric strain 3 s(V) with s(V) = (V/V0)^(1/3) - 1. The cubic result minimizes W over every symmetric strain of that trace; the uniform-scaling result uses e = s(V) delta without shape relaxation.", "By evaluation of the definitions (support lemmas, not cited), the Voigt contractions of an isotropic tensor are the Lame values K = lam + 2 mu/3 and G = mu, its Poisson ratio and Young's modulus are lam/(2(lam + mu)) and mu(3 lam + 2 mu)/(lam + mu), and the contractions of a cubic tensor are (C11 + 2 C12)/3 and (C11 - C12 + 3 C44)/5." ], "conventions": { "model_scope": "Linear elasticity of a homogeneous solid with a constant stiffness tensor. Physlib has no elastic-solid model, so these are local theorems about the OMAI formulas over Mathlib reals; no Physlib declaration is identified.", "component_scope": "Full fourth-rank tensor on Cartesian indices 1..3; isotropic and cubic symmetry only. Strains are symmetric; the deviatoric and volumetric parts are taken with respect to the Kronecker delta.", "normalization_basis": "Moduli are intensive, per unit volume of the solid. The equation-of-state energy is per cell of reference volume V0.", "frequency_convention": "Not applicable; no frequencies enter these formulas.", "unit_embedding": "Coherent SI: stiffness constants and moduli in Pa, energy in J, volume in m^3; nu is dimensionless. Mixed GPa and eV/A^3 inputs must be converted to one embedding first.", "held_fixed": "Temperature, composition and internal coordinates are fixed; the stiffness is the clamped-ion or relaxed-ion tensor as supplied and is not recomputed under strain. In the relaxed equation-of-state result the cell shape relaxes at each volume; in the uniform-scaling result it does not.", "admissibility": "Positive strain energy for every nonzero symmetric strain for the general bounds; C12 <= C11 and C44 >= 0 for the relaxed cubic result; V0 > 0 for both equation-of-state results.", "reference_energy": "E0 is an arbitrary constant offset of the cell energy and drops out of the curvature.", "qpoint_weighting": "Not applicable; no Brillouin-zone sums enter these formulas." }, "remaining_obligations": [ "DFT or finite-difference producers of ElasticConstants (compute_elastic_constants) are not verified, including the strain step, the stress sign convention, ionic relaxation and the symmetrization of the computed tensor.", "The Voigt averages are upper bounds for a general crystal and the Reuss averages lower bounds; neither bound is proved here, and the Hill value is an average of the two, not a bound. For a cubic crystal the Voigt bulk modulus is exact, but the Voigt shear modulus equals the Reuss value only at Zener ratio 2 C44 / (C11 - C12) = 1.", "Fitted equations of state (Birch-Murnaghan, Vinet, polynomial) and numerical second derivatives used by compute_bulk_modulus_eos are not verified; the equality with the Voigt modulus is shown only for linear-elastic cell energies at V0. For a general crystal the shape-relaxed curvature is the Reuss modulus, which is not proved here.", "Pressure-dependent moduli, finite-strain elasticity beyond the stated volumetric path, temperature dependence and anisotropic Poisson ratios are not covered." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIElasticity", "declarations": [ "OMAI.PhyslibBridge.Elasticity.voigt_pos_of_posDef", "OMAI.PhyslibBridge.Elasticity.poissonRatio_bounds_of_posDef", "OMAI.PhyslibBridge.Elasticity.isotropic_stable_iff", "OMAI.PhyslibBridge.Elasticity.cubic_relaxed_eosBulk_eq_voigtBulk", "OMAI.PhyslibBridge.Elasticity.uniformScaling_eosBulk_eq_voigtBulk" ], "source_path": "lean/bridge/OMAIElasticity.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_bulk_modulus_eos" }, { "kind": "operator", "id": "contract_bulk_modulus" }, { "kind": "operator", "id": "contract_poisson_ratio" }, { "kind": "operator", "id": "contract_shear_modulus" }, { "kind": "operator", "id": "contract_youngs_modulus" }, { "kind": "node", "id": "BulkModulus" }, { "kind": "node", "id": "PoissonRatio" }, { "kind": "node", "id": "ShearModulus" }, { "kind": "node", "id": "YoungsModulus" } ], "source_sha256": "dbc0efc3f35bcc223d28cd60d70fa591ab5f4cf9d285fe97360be6dca0602844" } }, { "id": "linearized-bte-conductivity", "relation": "local", "targets": [ { "kind": "node", "id": "MeanFreeDisplacement[bte_solver=direct_inverse]", "uid": "b24a26725f5641ffb16720514a7fdb23bf03304ce0d51c9f39501d1c8c009b6c" }, { "kind": "node", "id": "ThermalConductivity[bte_solver=direct_inverse]", "uid": "a369d33e9643b9f8766b111cde72c44fbf34a13ae46dfab31e1afab1c2129ca9" }, { "kind": "operator", "id": "contract_kappa[bte_solver=direct_inverse]", "uid": "8bd0005f31f01cd5d358f7cc58b21f7c7bb12abda8f11140d1b18c2b9b2e1e70", "formula_sha256": "2e54bbf0cc370a56c0ffd43f2ca8cdefa185eb02d8d2388124e6586960adac5c" }, { "kind": "operator", "id": "solve_bte[bte_solver=direct_inverse]", "uid": "3e5e0cdbda5333384f4df3d6b4e517854038fea4377f64e7fed212b9d995d2ea", "formula_sha256": "8ee139deb4c747f5b98fc57ede932ac9471296bb7fce4f9d04cd4d6dc0feafcc" } ], "statement": "The catalog's collision matrix Omega~ for the direct linearized BTE is symmetric for an even smearing kernel and a |V_3|^2 symmetric under exchange of its first two modes on the momentum-conserving triples (q, q', -q - q') the matrix reads, and so is its weighting M = C^(1/2) Omega~ C^(1/2). Without the in-scattering kernel the catalog's linear system M F = c v is solved exactly by the relaxation-time F = v/(2 Gamma), for nonzero linewidths and positive heat capacities. For a symmetric collision matrix the conductivity kappa^{alpha beta} = Sum c v^alpha F^beta / (V N_q) of any solution is a symmetric tensor, and it is positive semidefinite when Omega~ is and every mode heat capacity is nonnegative. For an even smearing kernel of the frequency offset, x^T Omega~ x = Sum 2 (Gamma - Gamma^db) x^2 plus a sum of squares over the momentum-conserving triples, where the detailed-balance linewidth Gamma^db weights each process by s' s''/s with s = sqrt(n(n + 1)), so Omega~ is positive semidefinite whenever Gamma >= Gamma^db. With exact energy conservation the catalog's three-phonon linewidth equals Gamma^db. With the nonnegative isotope and boundary linewidths added by Matthiessen's rule, Omega~ and the conductivity of the catalog chain are then positive semidefinite with no positivity hypothesis. The exact identity, not positivity alone, fixes the kernel constant pi/(N_q hbar^2) and the weight 2 sqrt(n''(n'' + 1)) relative to Gamma.", "sources": [], "conditions": [ "Every linewidth is the canonical Gamma = Im Sigma = 1/(2 tau). The Kronecker term of Omega~ is the scattering rate 2 Gamma = 1/tau, and its kernel carries pi/(N_q hbar^2), the constant of 2 Gamma.", "The mesh is a finite Gamma-centred grid of N_q points closed under q + q' + q'' = 0, and the smearing kernel is even, delta(-x) = delta(x). The exchange symmetry |V_3|^2(q nu, q' nu', -q - q' nu'') = |V_3|^2(q' nu', q nu, -q - q' nu'') holds in the collision labeling, where |V_3|^2(q, q', -q - q') is the catalog's decay-labeled kernel at (q, -q', q + q'). It follows from the translational invariance of the third-order force constants, momentum conservation and the time-reversal gauge e(-q) = e(q)*. The catalog's decay-labeled symbol is symmetric in its second and third modes instead. The positivity results also use that partner symmetry, which in the collision labeling is the exchange of the second and third modes, so |V_3|^2 is symmetric under every permutation of its three modes.", "The sum of squares needs a nonnegative |V_3|^2 and smearing kernel, and positive frequencies on the three modes of every triple that |V_3|^2 couples. A zero-frequency mode, such as an acoustic mode at q = 0, then decouples, as kALDo restricts the scattering matrix to physical_mode and phono3py computes |V_3|^2 only when all three frequencies exceed cutoff_frequency. Chaput's positive semidefiniteness (Phys. Rev. Lett. 110, 265506 (2013)) needs exact energy conservation, a kernel that vanishes at every nonzero offset. There the Bose identities 1 + n' + n'' = s' s''/s on the decay shell and n' - n'' = s' s''/s on the absorption shell make the catalog's three-phonon linewidth equal to Gamma^db, with omega(-q) = omega(q). With Gaussian smearing Gamma can fall below Gamma^db, and issue 221 gives a numerical Omega~ with a negative eigenvalue. phono3py's pinv_method=1 drops such eigenvalues from the pseudo-inverse. The tetrahedron method is not a kernel of the offset and is not covered.", "The three positive channels in the q + q' + q'' = 0 labeling equal the physical collision operator on deviations odd under q -> -q, where the drive c v lies. On even deviations they differ.", "The relaxation-time limit drops the in-scattering kernel, |V_3|^2 = 0 on the triples Omega~ reads, and keeps Gamma as an independent input, the total linewidth, with Gamma != 0 and c > 0 for every mode. The positivity results need V_cell N_q > 0 and c >= 0, and no sign of the individual modal contributions c v F." ], "conventions": { "model_scope": "The direct linearized phonon BTE and its conductivity contraction on a finite mesh. Physlib has no phonon-transport model, so these are local theorems about the OMAI formulas over Mathlib reals and matrices. No Physlib declaration is identified.", "component_scope": "The matrices Omega~ and M over all modes of the mesh, the three Cartesian components of F, and the full 3 x 3 conductivity tensor.", "normalization_basis": "Per mode on a mesh of N_q points with 1/N_q on the Brillouin-zone sums, and kappa per cell volume V_cell.", "frequency_convention": "Angular frequencies and linewidths, Gamma = Im Sigma = 1/(2 tau). A code reporting linear THz or the rate 1/tau needs the factor 2 pi or 1/2 before comparison.", "unit_embedding": "Coherent SI: omega and Gamma in rad/s, Omega~ in 1/s, c in J/K per mode, v in m/s, F in m, V_cell in m^3 and kappa in W/(m K). The occupations are dimensionless.", "held_fixed": "The mesh, the frequencies, |V_3|^2, the linewidths, the heat capacities and the velocities are fixed inputs. No convergence in the mesh is asserted.", "admissibility": "An even smearing kernel and an exchange-symmetric |V_3|^2 on momentum-conserving triples for the symmetry results, Gamma != 0 and c > 0 for the relaxation-time limit, c >= 0 and V_cell N_q > 0 for the conductivity. Positivity of Omega~ needs a nonnegative kernel and |V_3|^2, permutation symmetry of |V_3|^2, positive frequencies on every coupled triple, and either Gamma >= Gamma^db or exact energy conservation with omega(-q) = omega(q).", "reference_energy": "Not applicable.", "qpoint_weighting": "Uniform 1/N_q over the mesh. Irreducible-wedge weights and the unfolding by crystal symmetry are not modeled." }, "remaining_obligations": [ "Off shell, with Gaussian smearing, the catalog's three-phonon linewidth can fall below Gamma^db and Omega~ can be indefinite. Positive semidefiniteness then rests on the smearing approaching exact energy conservation as the mesh refines, which is not proved here.", "The relabeling between the decay-labeled and the collision-labeled kernels and the permutation symmetry of |V_3|^2 are assumed, not proved from the Maradudin-Fein form.", "The kernel |V_3|^2, built from the edge's ForceConstants[order=3] and Eigenvectors inputs by the definition on compute_linewidth[channel=anharmonic_3ph], the linewidths, the heat capacities and the velocities come from external codes and are not verified, nor is the convergence of the mesh sums.", "The existence and uniqueness of a solution of M F = c v and the numerical linear solver are not covered." ], "evidence": { "kind": "compiled_bridge", "module": "OMAILinearizedBTE", "declarations": [ "OMAI.PhyslibBridge.LinearizedBTE.chaputCollision_isSymm", "OMAI.PhyslibBridge.LinearizedBTE.lbteMatrix_isSymm", "OMAI.PhyslibBridge.LinearizedBTE.lbteSolves_rta_limit", "OMAI.PhyslibBridge.LinearizedBTE.lbteKappa_isSymm", "OMAI.PhyslibBridge.LinearizedBTE.lbteKappa_posSemidef", "OMAI.PhyslibBridge.LinearizedBTE.chaputCollision_quadratic", "OMAI.PhyslibBridge.LinearizedBTE.chaputCollision_posSemidef", "OMAI.PhyslibBridge.LinearizedBTE.anharmonicLinewidth_eq_detailedBalance", "OMAI.PhyslibBridge.LinearizedBTE.chaputCollision_posSemidef_onShell", "OMAI.PhyslibBridge.LinearizedBTE.lbteKappa_posSemidef_onShell" ], "source_path": "lean/bridge/OMAILinearizedBTE.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "contract_kappa[bte_solver=direct_inverse]" }, { "kind": "operator", "id": "solve_bte[bte_solver=direct_inverse]" }, { "kind": "node", "id": "MeanFreeDisplacement[bte_solver=direct_inverse]" }, { "kind": "node", "id": "ThermalConductivity[bte_solver=direct_inverse]" } ], "source_sha256": "5a3b595bd4894900071ef71ccf6d74a5b3ce1c2d2d5ff6782b10ee9276e87e23" } }, { "id": "md-correlation-estimators", "targets": [ { "kind": "node", "id": "HeatCurrentACF", "uid": "d8f44a4cce5e3dfea1ec9329189d49ac70e33d69cab222e2679f7791a30613ac" }, { "kind": "node", "id": "MeanSquaredDisplacement", "uid": "28ebe96bc049a92e4318fd5242778f30e5647531d0a2b39c5891546329c8bca0" }, { "kind": "node", "id": "ThermalConductivity[transport_model=green_kubo]", "uid": "25dc9f7af8aec52efb6be16b7b7c76f8334b03056f8838c3ff432fdad8482fb6" }, { "kind": "node", "id": "VelocityAutocorrelation", "uid": "f8b092359048e5130d56697ca9504bb4740b875aca685a8f760316bfb5e9a661" }, { "kind": "operator", "id": "autocorrelate_heat_current", "uid": "70b3a24785d80cd8cd1e3940c64032d39cad772ff92e73cd458f8b617ef40b1e", "formula_sha256": "44d85341be03b3ba6fccd6399e75dc62c09a61b36f4f3a2798f5d09b6d515a0e" }, { "kind": "operator", "id": "compute_msd", "uid": "ce9b639e5b88ed7e27cd2e0c879c620c582b0435d01365ea1dc35eb3fafdeeaa", "formula_sha256": "b4f9b442a0af59ae09e92327dd9753d2a819fddaa612827d64c5dfb7d03a7146" }, { "kind": "operator", "id": "compute_velocity_autocorrelation", "uid": "9c182752f0970df148341ff0e66df8e5be8c74744e79d9626847399355dab7a2", "formula_sha256": "f3c1b5e43f2f5fc7b9ee7b7b2f2202e3b9b282fec518c74a87f993d3ddb3c758" }, { "kind": "operator", "id": "contract_kappa[transport_model=green_kubo]", "uid": "89d2d05c87fbf2327a442b230ebc1349fbcc1ec158fb59ea5ab22f7727d915f7", "formula_sha256": "e2bb1307e05a6ec0ade5c2361ced5a02baf17d92a07791dad85ed9215e77ea16" } ], "relation": "local", "statement": "For a velocity record periodic over the origin window, v(t + n_lag) = v(t), with positions advanced by the stored velocity, r(t + 1) = r(t) + dt v(t), the catalog's mean squared displacement is M(tau) = dt^2 [tau Cv(0) + 2 sum_{k=1}^{tau-1} (tau - k) Cv(k)] with Cv the catalog's velocity autocorrelation: the Einstein and Green-Kubo diffusion estimators are one function of the trajectory. For a heat current periodic over the origin window with the Helfand moment advanced by the current, G(t + 1) = G(t) + dt J(t), and m >= 1, the symmetrized Green-Kubo conductivity (kappa_ab + kappa_ba)/2 at tau_max = m dt, with the integral of the sampled autocorrelation taken as the integral of its piecewise-linear interpolant (the trapezoid rule), equals the central-difference Einstein-Helfand estimator V (Lambda_ab(m + 1) - Lambda_ab(m - 1))/(4 k_B T^2 dt) with Lambda_ab(tau) = sum_t (G_a(t + tau) - G_a(t))(G_b(t + tau) - G_b(t))/n_lag. For a periodic heat current the catalog's origin-averaged autocorrelation is the inverse discrete Fourier transform of the cross spectrum conj(F J_a) F J_b/n_lag of one period, the FFT route.", "sources": [], "conditions": [ "compute_velocity_autocorrelation, compute_msd, autocorrelate_heat_current and contract_kappa[transport_model=green_kubo] are transcribed: origin sums over t = 1, ..., n_lag divided by N_atoms n_lag or n_lag, with the three Cartesian components summed into Cv and M, and kappa_ab = V int_0^tau_max Jcorr_ab dtau/(k_B T^2). Records are indexed by the integer sample; dt is the sampling step.", "Periodicity over the origin window, v(t + n_lag) = v(t) and J(t + n_lag) = J(t), is the circular-correlation hypothesis of an unpadded FFT of one period; the catalog's direct origin sums are linear correlations, which a zero-padded FFT reproduces. A trajectory of n_lag + tau_max samples correlated linearly is not periodic; the identities are exact for the circular estimators, and the end corrections of the linear estimator are not bounded here. Nothing is claimed about ensemble averages, ergodicity, finite-trajectory bias, statistical error or convergence in n_lag.", "r(t + 1) = r(t) + dt v(t) holds exactly when the stored velocity is the finite-difference velocity of unwrapped positions (the catalog's unwrap_pbc); for velocity-Verlet velocities it holds up to the dt^2 F/(2 m) term of the integrator step, which the identity does not absorb. G is the time-integrated catalog current, G(t + 1) = G(t) + dt J(t), the energy moment per unit volume (Helfand).", "The Green-Kubo integral from 0 to m dt is defined through the piecewise-linear interpolant of the samples Jcorr(k), k = 0, ..., m, so int = dt [Jcorr(0)/2 + sum_{k=1}^{m-1} Jcorr(k) + Jcorr(m)/2]; Simpson or rectangle rules differ at O(dt^2). The symmetrization (ab + ba)/2 is part of the identity: Jcorr_ab alone differs from the Helfand estimator by its antisymmetric part, and for a = b no symmetrization is needed. The identity holds for every real dt, k_B and T (both sides vanish at zero), so the physical ranges dt > 0, k_B > 0 and T > 0 are not hypotheses of the theorem.", "The FFT identity uses the samples J(1), ..., J(n_lag) of one period on Z/n_lag, F the unnormalized transform F x(k) = sum_j x_j exp(-2 pi i j k/n_lag) and F^-1 with the factor 1/n_lag; the cross spectrum is divided by n_lag again, matching the catalog's 1/n_lag.", "Units: v in m/s, r in m, dt in s, J in W/m^2, Jcorr in W^2/m^4, V in m^3, k_B in J/K, T in K, kappa in W/(m K). The catalog's ps, Angstrom and THz are conversions outside the theorems. compute_heat_current is not covered." ], "conventions": { "model_scope": "Discrete time-correlation estimators of a sampled classical trajectory: origin-averaged correlations, the Einstein and Helfand moments and the trapezoid Green-Kubo integral. Physlib has no molecular-dynamics trajectory model, so these are local theorems over Mathlib that identify no Physlib declaration.", "component_scope": "Scalar Cv and M summed over the three components and all atoms; the full 3x3 tensors Jcorr_ab, kappa_ab and Lambda_ab, with the symmetrized pair (ab, ba) for the Green-Kubo identity.", "normalization_basis": "Per atom and per origin for Cv and M; per origin for Jcorr and Lambda; kappa per unit volume through V and the current per unit volume.", "frequency_convention": "Not applicable; lags are integer sample indices and tau_max = m dt.", "unit_embedding": "Coherent SI as stated in the conditions; the theorems are exact-real identities and certify no floating-point or FFT implementation.", "held_fixed": "n_lag, dt, V, T and k_B are fixed inputs; the records are fixed functions of the sample index.", "admissibility": "Records periodic with period n_lag; integrated records advanced by dt times the rate; m >= 1 for the Green-Kubo identity; N_atoms and n_lag enter only as divisors.", "reference_energy": "Not applicable; per-atom energies enter only through the heat current, which is not covered.", "qpoint_weighting": "Not applicable; no Brillouin-zone sum enters." }, "remaining_obligations": [ "compute_heat_current is not covered: the per-atom virial and the Irving-Kirkwood, Hardy and virial decompositions of the energy current are not formalized.", "Ensemble averaging, ergodicity, the statistical error of a finite trajectory, the choice of tau_max (the plateau), finite-size effects, quantum corrections and the end effects of the linear (zero-padded) correlator are not addressed.", "The diffusivity from the slope of the mean squared displacement (contract_diffusivity) is a definition and is not bound; the Einstein-Helfand estimator under periodic boundaries with wrapped coordinates is not covered." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIMDCorrelations", "declarations": [ "OMAI.PhyslibBridge.MDCorrelations.catalogMSD_eq_vacf", "OMAI.PhyslibBridge.MDCorrelations.greenKubo_symm_eq_helfand", "OMAI.PhyslibBridge.MDCorrelations.catalogHeatACF_eq_invDFT" ], "source_path": "lean/bridge/OMAIMDCorrelations.lean", "build_record": "lean/bridge/build-evidence.json", "source_sha256": "5ad1d4f7c88f8483525123b63bf7b73a15ea0ed01363b14360a009e0189fc8cc" } }, { "id": "molecular-normal-modes", "relation": "local", "targets": [ { "kind": "node", "id": "MolecularFrequency", "uid": "e1270ea9a293c9107226bf831f60985f5525a0c386dca70841743bdc5d082c26" }, { "kind": "operator", "id": "compute_molecular_frequencies", "uid": "e8e86bed52c1e09311b9baada0501b7668b72cf8de3797735bc4c7bb60090f3f", "formula_sha256": "1b2f3949e74ab2b2db0136deee0dd84f2f576b76d18134a7624ebba18964fce0" } ], "statement": "Interpreting the opaque catalog frequencies as the eigenvalues of the mass-weighted Hessian of the potential at the structure, signed so that an imaginary mode is negative: at a local minimum of a potential twice continuously differentiable there, with positive masses, every eigenvalue is nonnegative, so no frequency is serialized negative (n_imaginary = 0) and each squared frequency is its eigenvalue. For a translation-invariant potential twice continuously differentiable at the structure the three mass-weighted rigid translations are zero modes. The catalog's projected list of 3N-6 or 3N-5 frequencies is a subset of these 3N values, so the sign result covers it.", "sources": [], "conditions": [ "The interpretation fixes the opaque producer as the harmonic normal-mode analysis of the potential at the given structure, with frequencies in coherent angular units; the catalog's cm^-1 wavenumbers are omega / (2 pi c).", "No negative eigenvalue at a local minimum; the converse fails, and the transition-state diagnostic n_imaginary = 1 is not covered.", "Masses are positive and the potential is twice continuously differentiable at the structure." ], "conventions": { "model_scope": "Harmonic normal modes of one molecular potential energy surface at a structure; a local theorem over Mathlib with no Physlib model.", "component_scope": "All 3N Cartesian coordinates, before any projection of translations or rotations.", "normalization_basis": "Mass-weighted coordinates sqrt(m) u.", "frequency_convention": "Angular frequency omega with omega^2 the Hessian eigenvalue; catalog wavenumbers are omega / (2 pi c) in cm^-1, and imaginary modes are serialized negative.", "unit_embedding": "Coherent SI: Hessian in J/m^2, masses in kg, omega in rad/s.", "held_fixed": "The structure; no temperature or environment enters.", "admissibility": "Local minimum of the potential, twice continuously differentiable there, positive masses; translation invariance for the zero modes.", "reference_energy": "An additive constant in the potential changes no mode.", "qpoint_weighting": "Not applicable; molecules have no Brillouin zone." }, "remaining_obligations": [ "Rotational zero modes at a stationary point, the 3N-6 or 3N-5 count of vibrations and the projection of translations and rotations that quantum-chemistry codes apply are not covered.", "Numerical Hessians (finite differences, analytic derivatives with incomplete convergence), anharmonicity and frequency scaling factors are not verified." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIMolecularModes", "declarations": [ "OMAI.PhyslibBridge.MolecularModes.molecularFrequencies_of_isLocalMin", "OMAI.PhyslibBridge.MolecularModes.massWeightedHessian_translation" ], "source_path": "lean/bridge/OMAIMolecularModes.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_molecular_frequencies" }, { "kind": "node", "id": "MolecularFrequency" } ], "source_sha256": "eea0883585af4e1ca7d8bc19613dcbfb014bff335ec4d3c0ffc75acdbc5214ad" } }, { "id": "participation-ratio-bounds", "relation": "local", "statement": "For a normalized vibrational mode on N >= 1 atoms, with per-atom amplitudes a_i = sum_alpha |e_{i,alpha}|^2, the OMAI participation ratio p = 1/(N sum_i a_i^2) satisfies 1/N <= p <= 1, with p = 1 exactly when a_i = 1/N on every atom and p = 1/N exactly when the mode lives on a single atom.", "targets": [ { "kind": "node", "id": "ParticipationRatio", "uid": "09138ae29cee41ae5f1a1758ac65a479625ef4d3c6026bf7ad5b60f884d668a0" }, { "kind": "operator", "id": "compute_participation_ratio", "uid": "93c6afe349f877656a16d18074e195ac4257226ce9d863d7f5d10ce96b679715", "formula_sha256": "5f3e67dbd9381917a0ab1c443fbf2698a741f2b3ce881210caac47777a0d07a9" } ], "sources": [], "conditions": [ "The eigenvector is normalized, sum_i sum_alpha |e_{i,alpha}|^2 = 1, over N >= 1 atoms and three Cartesian components, and a_i is the auxiliary per-atom amplitude of the catalog.", "The amplitudes are those of the mass-weighted eigenvector (the kALDo convention). In a multi-species crystal this differs from the displacement-based Bell-Dean participation ratio, which weights the amplitudes by 1/sqrt(M).", "The ratio measures the spread of one mode over the N basis atoms at fixed q, not its spatial extent in a supercell. The bounds use only the normalization and hold for any normalized vector." ], "conventions": { "model_scope": "Mathlib-only result about the participation ratio. Physlib has no vibrational-localization model, so no Physlib declaration is identified.", "component_scope": "One mode at fixed q; per-atom amplitudes summed over the three Cartesian components.", "normalization_basis": "1/N normalization over the N basis atoms; the eigenvector is normalized to one.", "frequency_convention": "Not applicable; no frequency enters the formula.", "unit_embedding": "Dimensionless.", "held_fixed": "The mode, its wavevector and its eigenvector.", "admissibility": "N >= 1 and a normalized mass-weighted eigenvector.", "reference_energy": "Not applicable.", "qpoint_weighting": "Not applicable; the ratio is per mode at fixed q." }, "remaining_obligations": [ "Eigenvector producers (phonopy, kaldo) and their normalization are not verified; an unnormalized eigenvector shifts p by the squared norm.", "Localization thresholds and the scaling of p with system size used to classify extended, diffusive and localized modes are not addressed." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibParticipationRatio", "declarations": [ "OMAI.PhyslibBridge.Lattice.participationRatio_bounds", "OMAI.PhyslibBridge.Lattice.participationRatio_eq_one_iff", "OMAI.PhyslibBridge.Lattice.participationRatio_eq_inv_iff" ], "source_path": "lean/bridge/OMAIPhyslibParticipationRatio.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_participation_ratio" }, { "kind": "node", "id": "ParticipationRatio" } ], "source_sha256": "2306537ae007125fc41ba3654b265cb1fd19cca3ab33769cf1590bad269203fa" } }, { "id": "phonon-scattering-channels", "relation": "local", "targets": [ { "kind": "node", "id": "Linewidth[channel=anharmonic_3ph]", "uid": "f52d161ad0be284c121c6fd10fe403c922aaa59646746c12719622004da5fa3a" }, { "kind": "node", "id": "Linewidth[channel=boundary]", "uid": "81288248a2b40f0abfe78b058ffa4025a0312324e8af26b801017b9a0971961e" }, { "kind": "node", "id": "Linewidth[channel=isotope]", "uid": "6cbd3322c5b6eb0105f9ca22d9ddc88334206ec9c115a31e6fb87409cc374c45" }, { "kind": "node", "id": "Linewidth[channel=total]", "uid": "50c33a1edd60bff6fb74124af1a8847a4c8a2c58d39554b4661e6a2cddb62f30" }, { "kind": "node", "id": "MeanFreeDisplacement[bte_solver=rta]", "uid": "ad1373f31b9dbb8bd24ea16e1f90f1d92a39d309f2dc22583fad4fd82eca6f59" }, { "kind": "operator", "id": "compute_boundary_scattering", "uid": "1326da3355b506b5142329e3b074a09589b95e4e18d1819c4c44ad579c1fd41e", "formula_sha256": "4710be932ea5cc929dedfe2bb5cf1b7e240e1781fa12cc45f9dcdd33c6c8c78e" }, { "kind": "operator", "id": "compute_isotope_scattering", "uid": "7cf53f521738578b03e17d568ddf5db3df454b0a47bd4d621390c5c9250dec73", "formula_sha256": "bbcc0cd3146146e003b4319f36af2916f769fadd9449a39309e565afc2b97b21" }, { "kind": "operator", "id": "compute_linewidth[channel=anharmonic_3ph]", "uid": "bed23822fc14b9f811e9d35b8361d56e0e38a34327b9e5b2c4e7f5aad6eebb72", "formula_sha256": "c5b1c700e285f83bade4999b264e709ae736be2f329f4eabb6ab45a86980b7e4" }, { "kind": "operator", "id": "solve_bte[bte_solver=rta]", "uid": "d1c7da072f801890c5b4d17bff8f196b38823ae46102eb308c2c50438edf8b19", "formula_sha256": "0b94f07d722c2b4c036e62b1790aa56e7427ffdd433bb5458e23219e6d9b923c" }, { "kind": "operator", "id": "sum_linewidths", "uid": "895d9e39c50870b6929c98b38c37b813f0688f3ade6b09e72c1bf1fad683ab7c", "formula_sha256": "be92953f056d5d9606396776a4b777bcf1806a44aea1dcf0e82ea7527dcbf85b" } ], "statement": "The catalog's Tamura isotope linewidth (pi/4N_q) omega^2 Sum_{q'nu'} delta(omega - omega') Sum_b g_b |e_b(q nu)* . e_b(q' nu')|^2 is nonnegative for nonnegative mass variances and smearing kernel, and for normalized eigenvectors it is at most (pi/4) omega^2 g_max D(omega), with D(omega) = (1/N_q) Sum_{q'nu'} delta(omega - omega') the density of states smeared with the isotope kernel. The catalog's three-phonon linewidth (pi/2N_q hbar^2) Sum_{q'nu'nu''} |V_3|^2 [(1 + n' + n'') delta(omega - omega' - omega'') + 2(n' - n'') delta(omega + omega' - omega'')] is nonnegative for a |V_3|^2 symmetric under exchange of the two partner phonons, as the Maradudin-Fein form is, and an even smearing kernel that does not increase with |x|, which covers the catalog's Gaussian. For an arbitrary |V_3|^2 it is nonnegative when the kernel vanishes at signed offsets above omega_q nu. A compiled example shows that an even kernel that is not monotone in |x| can make it negative even with the exchange symmetry. With the Casimir boundary linewidth |v|/(2L) in Matthiessen's sum and the other channels nonnegative, the relaxation-time mean free displacement F = v/(2 Gamma_total) of a moving mode has |F| <= L, with equality exactly when the anharmonic and isotope channels vanish. The bound holds because the boundary linewidth and the relaxation time carry the same factor 2 of Gamma = 1/(2 tau). A boundary rate written as |v|/L in a linewidth slot would cap |F| at L/2.", "sources": [], "conditions": [ "Every linewidth is the canonical Gamma = Im Sigma = 1/(2 tau) in angular units. The rates 1/tau that ShengBTE and kALDo report are twice it.", "The mesh is a Gamma-centred grid of N_q points closed under q - q', and every Brillouin-zone sum carries 1/N_q. The isotope bound uses eigenvectors normalized over the basis atoms and Cartesian axes of the cell, Sum_b Sum_alpha |e|^2 = 1, and mass-variance factors g_b in [0, g_max]. D(omega) integrates to the number of branches, 3N, not to 1.", "The three-phonon results need the mode's own frequency positive and a positive frequency for every partner that |V_3|^2 couples to. The acoustic modes at Gamma, where the occupation diverges, are admissible where |V_3|^2 vanishes on them, which the translational invariance of the force constants gives in the limit and codes impose by skipping zero frequencies.", "The exchange symmetry |V_3|^2(q nu, q' nu', q - q' nu'') = |V_3|^2(q nu, q - q' nu'', q' nu') follows from the permutation symmetry of the third-order force constants. Without it, or for a kernel that is not monotone in |x|, a negative absorption term needs kernel weight at a signed offset omega + omega' - omega'' above omega_q nu.", "The Casimir bound needs L > 0 and a nonzero group velocity, and holds for any nonnegative anharmonic and isotope linewidths, not only the catalog's." ], "conventions": { "model_scope": "Phonon scattering channels and the relaxation-time mean free path on a finite Brillouin-zone mesh. Physlib has no phonon-scattering model, so these are local theorems about the OMAI formulas over Mathlib reals and complex numbers. No Physlib declaration is identified.", "component_scope": "Per mode (q, nu): the three linewidth channels and their Matthiessen sum, and the three Cartesian components of the mean free displacement.", "normalization_basis": "Per mode on a mesh of N_q points with 1/N_q on every Brillouin-zone sum. Eigenvectors are normalized per cell over basis atoms and Cartesian axes.", "frequency_convention": "Angular frequencies and linewidths, Gamma = Im Sigma = 1/(2 tau). A code reporting linear THz or the rate 1/tau needs the factor 2 pi or 1/2 before comparison.", "unit_embedding": "Coherent SI: omega and Gamma in rad/s, hbar in J s, k_B in J/K, T in K, |V_3|^2 in J^2, v in m/s, L and F in m. g_b, the occupations and the eigenvector components are dimensionless.", "held_fixed": "The mesh, the frequencies, eigenvectors, |V_3|^2, temperature and boundary length are fixed inputs. No convergence in the mesh is asserted.", "admissibility": "g_b >= 0, |V_3|^2 >= 0, a nonnegative smearing kernel, positive frequencies for the three-phonon result, L > 0 and a nonzero velocity for the Casimir bound.", "reference_energy": "Not applicable.", "qpoint_weighting": "Uniform 1/N_q over the mesh. Irreducible-wedge weights are not modeled." }, "remaining_obligations": [ "The kernel |V_3|^2, the frequencies and eigenvectors and the mass variances come from external codes and are not verified, nor is the exchange symmetry of a code's |V_3|^2 after symmetrization and interpolation.", "Tetrahedron weights, which are not an even kernel in the offset, and the convergence of the mesh sums are not covered.", "The linearized-BTE solution, the collision matrix and the anharmonic and isotope in-scattering terms are not covered by this binding." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhononScattering", "declarations": [ "OMAI.PhyslibBridge.PhononScattering.isotopeLinewidth_nonneg", "OMAI.PhyslibBridge.PhononScattering.isotopeLinewidth_le_dos", "OMAI.PhyslibBridge.PhononScattering.anharmonicLinewidth_nonneg", "OMAI.PhyslibBridge.PhononScattering.anharmonicLinewidth_nonneg_of_support", "OMAI.PhyslibBridge.PhononScattering.anharmonicLinewidth_neg_of_nonmonotone", "OMAI.PhyslibBridge.PhononScattering.rta_casimir_bound" ], "source_path": "lean/bridge/OMAIPhononScattering.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "compute_boundary_scattering" }, { "kind": "operator", "id": "compute_isotope_scattering" }, { "kind": "operator", "id": "compute_linewidth[channel=anharmonic_3ph]" }, { "kind": "operator", "id": "solve_bte[bte_solver=rta]" }, { "kind": "operator", "id": "sum_linewidths" }, { "kind": "node", "id": "Linewidth[channel=anharmonic_3ph]" }, { "kind": "node", "id": "Linewidth[channel=boundary]" }, { "kind": "node", "id": "Linewidth[channel=isotope]" }, { "kind": "node", "id": "Linewidth[channel=total]" }, { "kind": "node", "id": "MeanFreeDisplacement[bte_solver=rta]" } ], "source_sha256": "a3fe99ec9df1ae2432a92534361bca2989b6ea8341b3d3f81648d41b28ec6860" } }, { "id": "static-dielectric-ionic-response", "targets": [ { "kind": "node", "id": "StaticDielectricTensor", "uid": "41f6c971e70bc42117a79b8ec51039f498280fdd9b77ef433bb602f90f966d93" }, { "kind": "operator", "id": "apply_nac_correction", "uid": "b8fd10813fea32fe974f69f9bf3b2129e374e4556047828e7ffb768249e5bb12", "formula_sha256": "f81d4dc1d53be67dc0902b57c7bf91ec745a39a8d584827ffd023470de9e431f" }, { "kind": "operator", "id": "compute_static_dielectric", "uid": "f658f73a798370ff39c363aef9b0d2219dd68d76251a561ee42d3b41e2d84a4a", "formula_sha256": "1aeb264ede74c9b48a916c554fbae4dfba2344232049591cd7231a254307fddd" } ], "relation": "local", "statement": "compute_static_dielectric is interpreted as eps_0 = eps_inf + c Z^T D^-1 Z, with D the Gamma-point dynamical matrix on the optical coordinates (positive definite once the acoustic translations are projected out of a stable crystal), Z the mass-weighted Born-charge matrix on those coordinates and c = 4 pi e^2/Omega (Gaussian; e^2/(epsilon_vac Omega) in SI, epsilon_vac the vacuum permittivity), which in the eigenbasis of D is the mode-oscillator-strength sum sum_m Z~_m Z~_m^T/omega_m^2. With D positive definite and c > 0, eps_0 - eps_inf is positive semidefinite and u (eps_0 - eps_inf) u = 0 exactly when Z u = 0, no polar mode couples to u. Along any direction u with u eps_inf u > 0, adding the catalog's non-analytic correction in its q -> 0 limit, the Gonze-Lee term c (Z u)(Z u)^T/(u eps_inf u), to D gives a Hermitian matrix whose product of eigenvalues (the longitudinal squared frequencies along u) divided by the product of the eigenvalues of D (the transverse squared frequencies) equals (u eps_0 u)/(u eps_inf u): the generalized Lyddane-Sachs-Teller relation.", "sources": [], "conditions": [ "compute_static_dielectric is an opaque output in the catalog, VASP's DFPT ionic tensor, the mode-strength sum over the Born charges and the Gamma-point phonons, added to the electronic tensor by atomate2. The interpretation needs the Gamma-point dynamical matrix D, the frequencies and eigenvectors the catalog lists as inputs together: eps_0 - eps_inf = c Z^T D^-1 Z equals the mode sum sum_m Z~_m Z~_m^T/omega_m^2 with Z~_m = Z^T e_m the mode-oscillator strength of the eigenvector e_m.", "D is the mass-weighted force-constant matrix at Gamma restricted to the 3N - 3 optical coordinates: the three rigid translations, whose omega^2 = 0 would make D singular, are projected out, and every optical omega^2 must be positive (a stable crystal with no imaginary Gamma mode). Z u has no component along the translations when the Born charges satisfy charge neutrality, sum_kappa Z*_kappa = 0; phonopy enforces this sum rule and the acoustic sum rule on the force constants, and both are assumed.", "Z_{kappa alpha, beta} = Z*_{kappa, beta alpha}/sqrt(M_kappa), beta the field direction and alpha the displacement, with Z* in units of e, and c carries e^2 and 1/Omega with Omega the primitive-cell volume, so c Z^T D^-1 Z is dimensionless: with D in s^-2, M in kg, Omega in m^3 and eps as a relative permittivity it is e^2/(eps_0 Omega) Z^T D^-1 Z.", "The Lyddane-Sachs-Teller statement uses the Gonze-Lee q -> 0 correction along u, D + c (Z u)(Z u)^T/(u eps_inf u), whose eigenvalues are the longitudinal squared frequencies along u; it holds for any c and any u with u eps_inf u > 0 and compares the products of all 3N - 3 optical squared frequencies, not individual LO-TO pairs. It is the q -> 0 limit of the catalog's apply_nac_correction along the direction u of q, with c = 4 pi q_e^2/Omega and Z the mass-weighted Born-charge matrix (apply_nac_correction is transcribed as nonAnalytic); the Gaussian-truncated Ewald sum at finite q is a representation choice outside the theorem." ], "conventions": { "model_scope": "Static lattice dielectric response of a harmonic insulator at Gamma, with the ionic term as the inverse of the optical dynamical matrix. Physlib has no dielectric or Born-charge model, so these are local theorems over Mathlib matrices that identify no Physlib declaration.", "component_scope": "The full 3x3 tensor eps_0 for positivity; one direction u at a time for the Lyddane-Sachs-Teller product.", "normalization_basis": "Per primitive cell: Omega is the cell volume and the Born charges and masses are per site of the cell.", "frequency_convention": "Angular squared frequencies omega^2 as the eigenvalues of D in s^-2; a dynamical matrix in linear THz^2 needs the factor (2 pi 10^12)^2 before use with SI c.", "unit_embedding": "eps_inf and eps_0 as relative permittivities, Z* in units of e, masses in kg, Omega in m^3, D in s^-2 and c = e^2/(epsilon_vac Omega) in SI, epsilon_vac the vacuum permittivity, or c = 4 pi e^2/Omega with Gaussian charges. Exact-real statements; floating-point evaluation is not certified.", "held_fixed": "eps_inf, the Born charges, the masses, the cell volume and the force constants are fixed inputs; temperature does not enter.", "admissibility": "D positive definite (optical coordinates of a stable crystal) and c > 0 for positivity; u eps_inf u > 0 for the Lyddane-Sachs-Teller direction.", "reference_energy": "Not applicable; no energy reference enters a dielectric tensor.", "qpoint_weighting": "Gamma only; no Brillouin-zone sum enters." }, "remaining_obligations": [ "The DFPT or finite-difference producers of eps_inf, Z* and the Gamma force constants, the enforcement of the charge-neutrality and acoustic sum rules and the projection of the acoustic modes are not verified.", "Finite-q non-analytic corrections (Gonze-Lee Ewald sums), the correction of individual LO frequencies and the frequency-dependent Lorentz-oscillator dielectric function are not covered; only the q -> 0 limit of apply_nac_correction is transcribed.", "Metals, free carriers and electronic screening beyond eps_inf, and anharmonic or temperature-dependent corrections to the static response are not covered." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIStaticDielectric", "declarations": [ "OMAI.PhyslibBridge.StaticDielectric.staticDielectric_sub_posSemidef", "OMAI.PhyslibBridge.StaticDielectric.lyddaneSachsTeller" ], "source_path": "lean/bridge/OMAIStaticDielectric.lean", "build_record": "lean/bridge/build-evidence.json", "source_sha256": "16a387817543a5bc9e530f3ef6d377909d0994670788d47ab5c90596c3317da8" } }, { "id": "thermodynamic-pressure", "targets": [ { "kind": "node", "id": "Pressure", "uid": "4ca6efdb874d146a6cca564c7a8a0504ee8868ae5890d4c33abddc147423d6f1" } ], "relation": "conceptual", "statement": "Physlib pressure is minus the volume derivative of Helmholtz free energy; OMAI pressure is a compressive stress trace, so their equality needs a thermodynamic stress interpretation.", "sources": [ { "module": "Physlib.StatisticalMechanics.MicroCanonicalEnsemble.ThermoQuantities", "name": "NVEHamiltonian.pressure", "kind": "def", "line": 219, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/StatisticalMechanics/MicroCanonicalEnsemble/ThermoQuantities.lean#L219" } ], "conditions": [ "Fixed composition and T, differentiable free energy in volume, compatible mechanical equilibrium and stress convention." ], "conventions": { "model_scope": "Equilibrium system described by an NVEHamiltonian family.", "component_scope": "Scalar pressure; no automatic equality to arbitrary stress components.", "normalization_basis": "not applicable", "frequency_convention": "not applicable", "unit_embedding": "MicroHamiltonian uses beta=1/T and F=-T log Z; derive physical kB/temperature scaling before SI comparison.", "held_fixed": "Temperature and particle/model parameters during V derivative.", "admissibility": "not applicable", "reference_energy": "Same free-energy reference and volume family.", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "Prove units and stress/free-energy conjugacy; existing source declaration is not the OMAI contract_pressure theorem." ], "evidence": { "kind": "source_correspondence" } }, { "id": "thermoelectric-figure-of-merit", "targets": [ { "kind": "node", "id": "ZT", "uid": "cf5879d2e17024fb396d83e19e71852f6cedf446fcf9d2feb81eb994e3e66e3a" }, { "kind": "operator", "id": "contract_zt", "uid": "ff8a1d876f02c91d2c28a2ebecd11217c9496adc0b928b06ff13847bb6ac427b", "formula_sha256": "261e8f1f0c68d1d4aca31e5984b597032ab4f2a3a3e0aa7b84beb577e095ae16" } ], "relation": "conditional", "statement": "OMAI zT = PF*T/kappa_total agrees with Physlib figureOfMerit in the explicitly scalar isotropic thermoelectric model.", "sources": [ { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.figureOfMerit", "kind": "def", "line": 158, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L158" }, { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.figureOfMerit_eq", "kind": "lemma", "line": 169, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L169" }, { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.figureOfMerit_pos", "kind": "lemma", "line": 182, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L182" }, { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.figureOfMerit_le_of_le", "kind": "lemma", "line": 200, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L200" } ], "conditions": [ "Valid material gives strictly positive total thermal conductivity.", "Positive zT requires T > 0 and S != 0.", "Lattice-only monotonicity requires positive replacement kappa_l, ordered lattice coefficients, T >= 0, and all other coefficients fixed." ], "conventions": { "model_scope": "Isotropic scalar thermoelectric material in linear response at one shared operating point.", "component_scope": "Scalar electronic conductivity and Seebeck; lattice and total conductivity use only an explicitly isotropic diagonal specialization, never arbitrary tensor entries or trace averaging.", "normalization_basis": "Intensive bulk transport coefficients, not device conductance.", "frequency_convention": "not applicable", "unit_embedding": "sigma in S/m; S in V/K; kappa in W/(m K); absolute T in K; PF in W/(m K^2); zT dimensionless.", "held_fixed": "All coefficients evaluated at the same T, carrier density/doping, composition, and model.", "admissibility": "sigma > 0, kappa_l > 0, kappa_e >= 0; coefficients must describe the material.", "reference_energy": "not applicable", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "The scalar/isotropic formula, positivity, and fixed-other-coefficients lattice monotonicity are compiled. Arbitrary anisotropic zT, numerical solver correctness, and causal effects of material interventions are not proved." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibSemantics", "declarations": [ "OMAI.PhyslibBridge.figureOfMerit_eq_physlib", "OMAI.PhyslibBridge.figureOfMerit_pos", "OMAI.PhyslibBridge.figureOfMerit_le_of_lattice_le", "OMAI.PhyslibBridge.isotropic_figureOfMerit_eq_physlib" ], "source_path": "lean/bridge/OMAIPhyslibSemantics.lean", "source_sha256": "9ce9bbe14c751333465a9f08bdb5319f369361e136f2f8233e8e72e6614797a5", "build_record": "lean/bridge/build-evidence.json" } }, { "id": "thermoelectric-input-coefficients", "targets": [ { "kind": "node", "id": "ElectricalConductivity[carrier=electronic]", "uid": "c4a81f649b3964aed4d4b3c48dfec4f252958c9560332286e2105b175794144b" }, { "kind": "node", "id": "ElectronicThermalConductivity", "uid": "960b38ae04ecc9fbaacb02d9ae832f15e84b8cac0563e7f6a62ddf32c61d725c" }, { "kind": "node", "id": "SeebeckCoefficient", "uid": "80778c5d6cb9f0d1dd23d5c6cd6059acea161892532ce7c581c7477257debf02" } ], "relation": "conditional", "statement": "The OMAI scalar electronic transport quantities can supply the sigma, S, and kappa_e fields of a valid Physlib thermoelectric material at the same operating point.", "sources": [ { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial", "kind": "structure", "line": 102, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L102" } ], "conditions": [ "sigma > 0 and kappa_e >= 0; S may have either sign.", "Provide an independently justified lattice conductivity kappa_l > 0." ], "conventions": { "model_scope": "Isotropic scalar thermoelectric material in linear response at one shared operating point.", "component_scope": "Scalar electronic conductivity and Seebeck; lattice and total conductivity use only an explicitly isotropic diagonal specialization, never arbitrary tensor entries or trace averaging.", "normalization_basis": "Intensive bulk transport coefficients, not device conductance.", "frequency_convention": "not applicable", "unit_embedding": "sigma in S/m; S in V/K; kappa in W/(m K); absolute T in K; PF in W/(m K^2); zT dimensionless.", "held_fixed": "All coefficients evaluated at the same T, carrier density/doping, composition, and model.", "admissibility": "sigma > 0, kappa_l > 0, kappa_e >= 0; coefficients must describe the material.", "reference_energy": "not applicable", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "Identify the representations, units and operating point, and verify the sign conditions; no upstream BTE or ab-initio producer theorem is identified." ], "evidence": { "kind": "source_correspondence" } }, { "id": "thermoelectric-power-factor", "targets": [ { "kind": "node", "id": "PowerFactor", "uid": "6c2382614be093f934d4f830419db9d60d32610041a8c213b677efc42f6a98ee" }, { "kind": "operator", "id": "contract_power_factor", "uid": "692d823ec06a56ba59453ff8650d602fc72d1aac991aae3e572b236e5a269306", "formula_sha256": "0b12a94e400c0db0bfa5be59bc424b1869508be637badb889fb29d42fd6b39c7" } ], "relation": "exact_definition", "statement": "Under the declared scalar field interpretation, OMAI PF = sigma_e*S^2 is Physlib ThermoelectricMaterial.powerFactor.", "sources": [ { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.powerFactor", "kind": "def", "line": 125, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L125" }, { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.powerFactor_pos", "kind": "lemma", "line": 128, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L128" } ], "conditions": [ "Use a ThermoelectricMaterial satisfying its field constraints; identify OMAI scalar sigma_e with M.sigma and Seebeck with M.S.", "Strict positivity additionally requires S != 0." ], "conventions": { "model_scope": "Isotropic scalar thermoelectric material in linear response at one shared operating point.", "component_scope": "Scalar electronic conductivity and Seebeck; lattice and total conductivity use only an explicitly isotropic diagonal specialization, never arbitrary tensor entries or trace averaging.", "normalization_basis": "Intensive bulk transport coefficients, not device conductance.", "frequency_convention": "not applicable", "unit_embedding": "sigma in S/m; S in V/K; kappa in W/(m K); absolute T in K; PF in W/(m K^2); zT dimensionless.", "held_fixed": "All coefficients evaluated at the same T, carrier density/doping, composition, and model.", "admissibility": "sigma > 0, kappa_l > 0, kappa_e >= 0; coefficients must describe the material.", "reference_energy": "not applicable", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "The reviewed scalar formula interpretation and positivity transfer are compiled. Numerical electronic transport producers, finite precision, and physical adequacy are not certified." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibSemantics", "declarations": [ "OMAI.PhyslibBridge.powerFactor_eq_physlib", "OMAI.PhyslibBridge.powerFactor_pos" ], "source_path": "lean/bridge/OMAIPhyslibSemantics.lean", "source_sha256": "9ce9bbe14c751333465a9f08bdb5319f369361e136f2f8233e8e72e6614797a5", "build_record": "lean/bridge/build-evidence.json" } }, { "id": "thermoelectric-total-conductivity", "targets": [ { "kind": "node", "id": "ThermalConductivity[bte_solver=direct_inverse]", "uid": "a369d33e9643b9f8766b111cde72c44fbf34a13ae46dfab31e1afab1c2129ca9" }, { "kind": "node", "id": "ThermalConductivity[contribution=total]", "uid": "2931c0e41adb40b9ce80f5169881f3a266a2e85d5b280223914fb3b61628aea9" }, { "kind": "operator", "id": "sum_thermal_conductivity", "uid": "e5d0388c6c18b3fb6cb08647ddaf431bfb39e60f33e0abd1ecdd2cb0c05575ee", "formula_sha256": "05612b74df513ca263a0cbb8a80f9c5f703dceced4703b2bb1f8adf76ff8f28e" } ], "relation": "conditional", "statement": "OMAI lattice-plus-electronic conductivity agrees with Physlib totalThermalConductivity only after an explicit isotropic scalar interpretation of the rank-two lattice and total nodes.", "sources": [ { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.totalThermalConductivity", "kind": "def", "line": 143, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L143" }, { "module": "Physlib.CondensedMatter.Thermoelectric.Basic", "name": "CondensedMatter.ThermoelectricMaterial.totalThermalConductivity_pos", "kind": "lemma", "line": 146, "source_url": "https://github.com/leanprover-community/physlib/blob/9b583cb6ef6e9cc537c4d4b786b508de1abbcf42/Physlib/CondensedMatter/Thermoelectric/Basic.lean#L146" } ], "conditions": [ "Lattice tensor is kappa_l times the identity; electronic scalar is interpreted as kappa_e times the identity.", "kappa_l > 0 and kappa_e >= 0; coefficients share operating point and normalization." ], "conventions": { "model_scope": "Isotropic scalar thermoelectric material in linear response at one shared operating point.", "component_scope": "Scalar electronic conductivity and Seebeck; lattice and total conductivity use only an explicitly isotropic diagonal specialization, never arbitrary tensor entries or trace averaging.", "normalization_basis": "Intensive bulk transport coefficients, not device conductance.", "frequency_convention": "not applicable", "unit_embedding": "sigma in S/m; S in V/K; kappa in W/(m K); absolute T in K; PF in W/(m K^2); zT dimensionless.", "held_fixed": "All coefficients evaluated at the same T, carrier density/doping, composition, and model.", "admissibility": "sigma > 0, kappa_l > 0, kappa_e >= 0; coefficients must describe the material.", "reference_energy": "not applicable", "qpoint_weighting": "not applicable" }, "remaining_obligations": [ "The scalar formula and explicitly isotropic tensor embedding are compiled. Identifying a specific simulation tensor as isotropic and proving BTE/adapter correctness remain outside this result." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIPhyslibSemantics", "declarations": [ "OMAI.PhyslibBridge.totalThermalConductivity_eq_physlib", "OMAI.PhyslibBridge.totalThermalConductivity_pos", "OMAI.PhyslibBridge.isotropic_total_conductivity" ], "source_path": "lean/bridge/OMAIPhyslibSemantics.lean", "source_sha256": "9ce9bbe14c751333465a9f08bdb5319f369361e136f2f8233e8e72e6614797a5", "build_record": "lean/bridge/build-evidence.json" } }, { "id": "thermoset-cure-kinetics", "relation": "local", "targets": [ { "kind": "node", "id": "CureConversion", "uid": "a0fea831928d4ff6617a0d102f74966d0ea4ce0a2e4cc160cbd362dd0fb8fc0a" }, { "kind": "node", "id": "GlassTransitionTemperature", "uid": "5789aa6f8439e8b71b48197dd92c6f608f2bf3ca9900bc631d2321f5eded13cb" }, { "kind": "operator", "id": "dibenedetto_glass_transition", "uid": "68e156cfe2da68f22c0e5c3c0bdc6426ce2fad2aa8a25714d90d79343d8d2ed4", "formula_sha256": "6ec6864e62dbe6e996b088bcc17ed2c238b3839a5ca2ba8a30b6cdbd4f3ea03a" }, { "kind": "operator", "id": "kamal_cure_conversion", "uid": "0b2784cfd1c1758e66988322d83589700be6f7b1184c6f36a8d073eda43b57e1", "formula_sha256": "f997bcdb5ce1ddd5b4061584d0e79463affd1ca1248131741721081bb20c7f00" } ], "statement": "The catalog's isothermal Kamal-Sourour cure time t(alpha) = int_{alpha_0}^{alpha} da/((1 - a)^n (k1 + k2 a^m)), k_i = A_i exp(-E_i/(N_A k_B T)), increases strictly with conversion below full cure and has derivative 1/rate. Near every interior cure time there is one inverse alpha(t), taking that time back to the conversion, that solves d alpha/dt = (k1 + k2 alpha^m)(1 - alpha)^n at every time of a neighbourhood, and every continuous local inverse solves the rate law. These are local statements, not a global solution of the rate law. The catalog's DiBenedetto relation maps conversions [0, 1] one to one and strictly increasingly onto [T_g0, T_ginf] for lambda > 0 and T_g0 < T_ginf, and for 0 < lambda < 1 and T_g0 < T_ginf it lies strictly below the linear interpolation at every interior conversion.", "sources": [], "conditions": [ "Isothermal cure at absolute temperature T in kelvin with A_1, A_2 >= 0 in 1/s, autocatalytic order m >= 0, any real order n, initial conversion alpha_0 >= 0, and either A_1 > 0 (which includes pure nth-order kinetics with A_2 = 0) or alpha_0 > 0 with A_2 > 0, so that the rate is positive from alpha_0 up to full cure; conversions lie in [alpha_0, 1).", "Conversion is a fraction in [0, 1], not a percentage. E_1 and E_2 are activation energies per mole and enter through R = N_A k_B; an activation energy per molecule must not be divided by N_A again.", "The rate law is the single-(1 - alpha)^n Kamal-Sourour form; diffusion control, a temperature-dependent maximum conversion and separate exponents on the two terms are other models. Isothermal Kamal values are meaningful only while T_g(alpha(t)) stays below the cure temperature, before vitrification.", "The DiBenedetto results use the Pascault-Williams form with lambda > 0 and alpha in [0, 1]; monotonicity, the bijection and the strict comparison with the linear interpolation require T_ginf > T_g0, and the comparison also 0 < lambda < 1. T_g0, T_ginf and T_g are in kelvin, as the GlassTransitionTemperature node and its evaluator store them; the Pascault-Williams algebra commutes with shifts of the temperature scale, which is a remark about the formula, not a change of storage unit.", "T_g(alpha) is assumed independent of the cure path and specific to one resin stoichiometry." ], "conventions": { "model_scope": "Phenomenological thermoset cure kinetics and the DiBenedetto glass-transition relation. Physlib has no polymer-cure model, so these are local theorems about the OMAI formulas over Mathlib reals; no Physlib declaration is identified.", "component_scope": "Scalar degree of cure and glass transition temperature of one homogeneous thermoset.", "normalization_basis": "Conversion is the fraction of reacted groups in [0, 1]; activation energies are per mole.", "frequency_convention": "Not applicable; rate constants are in 1/s and enter only through the Arrhenius factors.", "unit_embedding": "Coherent SI: T, T_g0, T_ginf and T_g in K, A_i in 1/s, E_i in J/mol, N_A in 1/mol, k_B in J/K, t_cure in s; alpha, m, n and lambda are dimensionless.", "held_fixed": "Temperature is constant during the cure; the kinetic parameters and the DiBenedetto parameters are fixed.", "admissibility": "T > 0 and the parameter ranges in the conditions; conversions at or above 1 and negative conversions have no asserted interpretation.", "reference_energy": "Not applicable.", "qpoint_weighting": "Not applicable." }, "remaining_obligations": [ "Fitted Kamal and DiBenedetto parameters and the measurements behind them are not verified.", "Vitrification and diffusion-controlled cure, non-isothermal schedules, cure-path dependence of T_g and the identification lambda = Delta C_p,inf / Delta C_p,0 are not covered.", "The rate law is solved only locally near each interior cure time; a global solution on [0, t(1-)) and the time to full cure, which diverges for n >= 1, are not proved." ], "evidence": { "kind": "compiled_bridge", "module": "OMAICureKinetics", "declarations": [ "OMAI.PhyslibBridge.CureKinetics.kamalCureTime_strictMonoOn", "OMAI.PhyslibBridge.CureKinetics.kamalCureTime_hasDerivAt", "OMAI.PhyslibBridge.CureKinetics.kamal_inverse_exists", "OMAI.PhyslibBridge.CureKinetics.kamal_inverse_solves_rate_law", "OMAI.PhyslibBridge.CureKinetics.diBenedetto_strictMonoOn", "OMAI.PhyslibBridge.CureKinetics.diBenedetto_bijOn", "OMAI.PhyslibBridge.CureKinetics.diBenedetto_lt_linear" ], "source_path": "lean/bridge/OMAICureKinetics.lean", "build_record": "lean/bridge/build-evidence.json", "proved_targets": [ { "kind": "operator", "id": "dibenedetto_glass_transition" }, { "kind": "operator", "id": "kamal_cure_conversion" }, { "kind": "node", "id": "CureConversion" }, { "kind": "node", "id": "GlassTransitionTemperature" } ], "source_sha256": "7575810b36eb5f97bdd12ef4d26e9c9b65e42f2a5e3913609e615e0d90980a2a" } }, { "id": "vacf-spectrum-lorentzian-limit", "targets": [ { "kind": "node", "id": "PhononDOS", "uid": "e14b6cdd007ca3af971aa52880811138606efdb47e30d5a26c2335903ce796f3" }, { "kind": "operator", "id": "fourier_to_dos", "uid": "c142a31f3e5f89612d68319b4c2186372bf15592f88b09a1f2cc1356564428c0", "formula_sha256": "b4cdd53d454584e5202fb019a24ae3f4bc8ef51ba843fd0210c4e91456c43c13" } ], "relation": "local", "statement": "For the velocity autocorrelation of lifetime-broadened modes, Cv(tau) = sum_m A_m exp(-Gamma_m tau) cos(omega_m tau) with every Gamma_m > 0, the catalog transform g(omega) = (6/(pi Cv(0))) int_0^infinity cos(omega tau) Cv(tau) dtau, with Cv(0) = sum_m A_m, equals sum_m (3 A_m/Cv(0))[L(Gamma_m, omega - omega_m) + L(Gamma_m, omega + omega_m)] with L(Gamma, x) = Gamma/(pi (Gamma^2 + x^2)) the normalized Lorentzian of half-width Gamma. For Cv(0) = sum_m A_m != 0 its integral is 6 over the whole real line and 3 over omega > 0 for every set of positive linewidths. As the linewidths s gamma_m shrink to zero, int g(omega) phi(omega) domega converges, for every integrable phi continuous at +-omega_m, to sum_m (3 A_m/Cv(0))(phi(omega_m) + phi(-omega_m)): the harmonic delta-sum density of states with mode weights 3 A_m/Cv(0), counted at +omega_m and -omega_m.", "sources": [], "conditions": [ "fourier_to_dos is transcribed as the Lebesgue integral over (0, infinity) at angular frequency omega; the damped-cosine form of Cv is a hypothesis on the input, not derived from the dynamics, and every linewidth must be positive (an undamped mode has no integrable transform).", "Normalization: the catalog's g is the density of states per atom, normalized to 3 states per atom on omega > 0: for Cv(0) = sum_m A_m != 0 it integrates to 6 over the whole real line and to 3 over omega > 0. Cv is the per-atom velocity autocorrelation summed over the three Cartesian components, so Cv(0) = 3 k_B T/m for one mass in classical equipartition, and the prefactor 6/(pi Cv(0)) removes its units. The (2/pi) int cos(omega tau) Cv(tau)/Cv(0) dtau convention normalizes the one-sided spectrum to 1 and differs by the factor 3. At Cv(0) = 0 the transform is 0 under the Lean convention x/0 = 0 and the integral values do not apply.", "Equal mode weights A_m are not implied; they hold for a classical harmonic crystal of one atomic mass in equipartition, and in general A_m are the velocity weights of the modes. The zero-linewidth limit is weak convergence against integrable test functions continuous at the mode frequencies, with all linewidths scaled by one factor s -> 0+.", "Frequencies are angular (rad/s) with tau in s; a spectrum on linear frequency in THz needs omega = 2 pi 10^12 nu. Windows, finite trajectories, the discrete cosine transform and the dos_broadening scheme of the catalog are outside the theorems." ], "conventions": { "model_scope": "Cosine transform of a sum of exponentially damped cosines and its zero-linewidth limit. Physlib has no molecular-dynamics or spectral-estimation model, so these are local theorems over Mathlib that identify no Physlib declaration.", "component_scope": "The scalar Cv summed over components and atoms, and its scalar transform g(omega) on the whole real line.", "normalization_basis": "g is normalized per atom: 3 states on omega > 0 and 6 over the real line, independent of the units of Cv.", "frequency_convention": "Angular frequency omega in rad/s and lag tau in s; Lorentzian half-widths Gamma_m in rad/s.", "unit_embedding": "Cv in m^2/s^2 per atom and g in s per atom (states per unit angular frequency; the prefactor 6/(pi Cv(0)) cancels the velocity units); exact-real statements that certify no numerical transform.", "held_fixed": "Mode amplitudes A_m, frequencies omega_m and linewidths Gamma_m are fixed inputs; the limit scales the linewidths by one factor.", "admissibility": "Every Gamma_m > 0 (or gamma_m > 0 with s > 0); phi integrable and continuous at +-omega_m for the limit; Cv(0) = sum_m A_m != 0 for the integral values 6 and 3.", "reference_energy": "Not applicable.", "qpoint_weighting": "Not applicable; modes enter as a finite list with their weights A_m." }, "remaining_obligations": [ "The damped-cosine form of the autocorrelation and the mode weights A_m are not derived from a trajectory or a harmonic model; lineshapes beyond one Lorentzian per mode are not covered.", "Finite trajectories, window functions, the discrete cosine transform and the Gaussian dos_broadening scheme are not addressed.", "The identification of the zero-linewidth limit with compute_dos on a q mesh (equal weights, the mode count 3 N) is not proved; only the weak limit to the weighted delta sum is." ], "evidence": { "kind": "compiled_bridge", "module": "OMAIVACFSpectrum", "declarations": [ "OMAI.PhyslibBridge.VACFSpectrum.catalogFourierDOS_damped", "OMAI.PhyslibBridge.VACFSpectrum.integral_catalogFourierDOS_damped", "OMAI.PhyslibBridge.VACFSpectrum.integral_Ioi_catalogFourierDOS_damped", "OMAI.PhyslibBridge.VACFSpectrum.tendsto_catalogFourierDOS_zero_linewidth" ], "source_path": "lean/bridge/OMAIVACFSpectrum.lean", "build_record": "lean/bridge/build-evidence.json", "source_sha256": "88e529acc3a8f224ce2e62aa781b8176694eeee06b49daa9fdd5d517ce32661d" } } ], "inventory": { "nodes": [ { "id": "ActivationEnergy", "uid": "5d35f8f6975629291c51d432d3b137fb5bd92165706d37dbab71366a4714b71d", "binding_ids": [ "arrhenius-ionic-transport" ], "status": "local", "reason": "No matching current Physlib Einstein-MSD, Arrhenius, Nernst-Einstein, or cluster-expansion formula theorem identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "AdsorptionEnergy", "uid": "949582fba59372799c672b38510bd07f2e733f8bd6e63887b9728a672b1c5b25", "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "AssessedDatabase", "uid": "3618484d27f4d34e4a8abf7db7b62201c5aefe00de89933375f082a883513bbd", "binding_ids": [], "status": "gap", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs." }, { "id": "AtomCount", "uid": "f1261195ad64911510ec9839d2e373c5cbda670f65a8a0c4437e37ff0a7838df", "binding_ids": [], "status": "gap", "reason": "No reviewed formula-level Physlib correspondence at this revision. The quantity may still occur elsewhere in Physlib." }, { "id": "AtomicMass", "uid": "12fb1ed0f109f6dd198199dde8ed169edc25c04ee12813e0612a96675784830c", "binding_ids": [], "status": "gap", "reason": "No reviewed formula-level Physlib correspondence at this revision. The quantity may still occur elsewhere in Physlib." }, { "id": "BandGap", "uid": "7164425d1695208e46171b5030c938c3545c180b57018d51748d0507f9f08135", "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "BareDynamicalMatrix", "uid": "e62e58716111fdc1dff24d18e75c70904cbb07ce6e0025e24fcdac8c4ed0a53d", "binding_ids": [ "crystal-dynamical-matrix" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "BondDissociationEnergy", "uid": "bf0f65e5f725c4d82321504f06db3610ce8e9d4d2041bf01dbcc9a1a02cfd191", "binding_ids": [], "status": "gap", "reason": "No current Physlib molecular Hessian, electronic gap, NEB, or bond-dissociation formula identification; harmonic oscillator analogies remain conceptual." }, { "id": "BornCharges", "uid": "438f0314dee5f9e98d88e4a7ba4425ef0edd207605338b318340da27bf28b2a4", "binding_ids": [], "status": "gap", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "BulkModulus", "uid": "c0a13e64c02811f49b951271c894373267937676e0c657fc9ef09fac72558ced", "binding_ids": [ "harmonic-qha-bulk-modulus", "harmonic-static-cp-identity", "harmonic-static-thermal-gruneisen-identity", "linear-elasticity-voigt-moduli" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "CalphadMolarEntropy", "uid": "b11c4ec0ddbfa4483cc0a65d6a8fb2cefa9687b9a86ebd11bfb968d4253ec022", "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "CarrierDensity", "uid": "e3ba3f3522daaaa0a91c35e0302be3bc11c00c176a6090863c45b3c2664fed37", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Einstein-MSD, Arrhenius, Nernst-Einstein, or cluster-expansion formula theorem identified." }, { "id": "CarrierMobility", "uid": "3303dee51d36b5a42bf80aa0ab62bdecd0db34cc77782c5203192ecf3512a91d", "binding_ids": [ "electronic-carrier-statistics" ], "status": "local", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "CellVolume", "uid": "cd81fd8a1d62a1484f3057647d23aa780500de44175ae4e686e189a3ca5b36b2", "binding_ids": [], "status": "gap", "reason": "No reviewed formula-level Physlib correspondence at this revision. The quantity may still occur elsewhere in Physlib." }, { "id": "ChemicalPotential", "uid": "e707090e106994d3718f0afe5a20f2e60037f543334fa79b4a00360b7962d527", "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ClassificationAccuracy", "uid": "0e10474b38987a0d5d3c77580649f461281a6a587093401a60781aab4bdf219d", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "ConductanceDriftExponent", "uid": "d9c306ec2d77f10a80335d58a2f12a09124ef4a0061ddf1ae18c90c3ef597aef", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "ConductanceRelaxationTime", "uid": "72bddad977642e53ed5cc05d9eec037d77a96a2c3232db16f5f4b4eaaff0b971", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "ConductanceState", "uid": "325f09352a8872995fe6ab0a40b9df8780a778389fa0fe00ebcbd9ef7758f4a3", "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ConductanceWindow", "uid": "ec3515d40f800b3c58242b644285b045dcfaec8102799a44dea2a61a9783673c", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "ConfigurationalEnergy", "uid": "12ec58944efc3849b267bf3017d9049179ed21717b7791d06fc17d69b7150c39", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Einstein-MSD, Arrhenius, Nernst-Einstein, or cluster-expansion formula theorem identified." }, { "id": "CumulativeKappa[wrt=mfp]", "uid": "b6a79d756cf45fc64a3a07feb9e46c7372f398b3ee54806bf1c9903d8a42039f", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "CumulativeKappa[wrt=omega]", "uid": "55eaa32f15db94d55448456fb26f1c3ecaafcc9a3da2947f1615258c492542ef", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "CureConversion", "uid": "a0fea831928d4ff6617a0d102f74966d0ea4ce0a2e4cc160cbd362dd0fb8fc0a", "binding_ids": [ "thermoset-cure-kinetics" ], "status": "local", "reason": "No matching current Physlib Kamal cure or DiBenedetto glass-transition model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "DepolarizationFactor", "uid": "297b3254a09037f017769b771e19ad849443911048f77078a804cbc334c38309", "binding_ids": [ "composite-effective-medium" ], "status": "local", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "DielectricTensor", "uid": "a84077d8f12cf490f370b5cd0b98a47e3e31b7ec75fc8763699763f8b2313db4", "binding_ids": [], "status": "gap", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "Diffusivity", "uid": "37eb727e4cf5e46c3ca04ff4bea4fe9dbb3f7f3c3fad8a1681c7e9d04bafd34f", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Einstein-MSD, Arrhenius, Nernst-Einstein, or cluster-expansion formula theorem identified." }, { "id": "DotProductError", "uid": "1e6467a5b03397510e0d7f33403fd1cb42de5ea579e1f36b074d8dbbc620da20", "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "DynamicalMatrix", "uid": "a883ea19a62fcb80dc79d8c347fc583bef6a56d8764aef89455da764b4732031", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "Eigenvectors", "uid": "21ba7756f45c808d3d91ed059c34c532f04845781d549b4c16573d9462bb2902", "binding_ids": [], "status": "gap", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ElasticConstants", "uid": "01043ab212fd87e4ba9493caf709ab9a7ccc86ec7a0047a0512af12eca1c391e", "binding_ids": [ "hyperelastic-elastic-constants" ], "status": "local", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ElectricalConductivity[carrier=electronic]", "uid": "c4a81f649b3964aed4d4b3c48dfec4f252958c9560332286e2105b175794144b", "binding_ids": [ "boltzmann-electronic-transport", "thermoelectric-input-coefficients" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ElectricalConductivity[carrier=ionic]", "uid": "3f5753096d30cc6741fe9c6fd5e247e9afbb87b94f5844ec233f9243345d3712", "binding_ids": [ "arrhenius-ionic-transport" ], "status": "local", "reason": "No matching current Physlib Einstein-MSD, Arrhenius, Nernst-Einstein, or cluster-expansion formula theorem identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ElectronAffinity", "uid": "0bc59239b7ed9882c822aae5818ec2c5449e656ab9897c32d45103b9d51a7b56", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "ElectronicDOS", "uid": "a32e08f3d5a147643986f961b701aea560791721df573f09188a557da369fb4f", "binding_ids": [ "electronic-carrier-statistics" ], "status": "local", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ElectronicThermalConductivity", "uid": "960b38ae04ecc9fbaacb02d9ae832f15e84b8cac0563e7f6a62ddf32c61d725c", "binding_ids": [ "boltzmann-electronic-transport", "thermoelectric-input-coefficients" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "EnergyAboveHull", "uid": "910d47d67a7308728e2f7d211a08c1590239c7b802cd75128f00cd103a719b84", "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Entropy", "uid": "5aa485c8eea2f4a2a9966f39589ad15e802327513f72c2366dc15605879b86ae", "binding_ids": [ "harmonic-mode-entropy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "FillerVolumeFraction", "uid": "093e6e3278c242894f90f40f3ef1d4b7fda19c58061ceb9e65ceddc2e0000c0d", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified." }, { "id": "ForceConstants[order=2]", "uid": "78d005a949ff56a526a1911f0b24e766daecad0a7f5ce9c1cc15222f2ed73551", "binding_ids": [ "force-constant-sum-rules" ], "status": "local", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source. The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ForceConstants[order=3]", "uid": "73372380af126dde147e403f69339b87ee2f938a896dd5b56e9100c055dc8f93", "binding_ids": [ "force-constant-sum-rules" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Forces", "uid": "9c5a2fdedab6b668ccf5f2043fb4c29a982bcde5063fd6ac8633ee2c7858ecbf", "binding_ids": [ "classical-harmonic-potential-force", "hellmann-feynman-force-balance" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "FormationEnergy", "uid": "a385b7ca981904ab5d82176557d453cea4995164adfa22a6ca195e1dfcd815d5", "binding_ids": [], "status": "gap", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified." }, { "id": "Frequency", "uid": "fa262448f625b62f5a8c6934518bb8ddc5d3a5a5ce69203b3e2f6b10b21cc4b8", "binding_ids": [ "classical-harmonic-frequency" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "GlassTransitionTemperature", "uid": "5789aa6f8439e8b71b48197dd92c6f608f2bf3ca9900bc631d2321f5eded13cb", "binding_ids": [ "thermoset-cure-kinetics" ], "status": "local", "reason": "No matching current Physlib Kamal cure or DiBenedetto glass-transition model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "GrainBoundaryEnergy", "uid": "f81c9c7907dc4297e44624098da25bfd87c1654a4198853bb185891282c24528", "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "GroupVelocity", "uid": "d1934e166dbd0c91125936f9b2ccc0100af0ba44ed439003aae76b47b6d26fc6", "binding_ids": [ "crystal-group-velocity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "Gruneisen", "uid": "a62ac1eae1e972a0b265a14f8a2af5610deb61eb0c883b7668ac4a64313b3b75", "binding_ids": [], "status": "gap", "reason": "The local quasiharmonic bridges cover the windowed Gibbs minimum, its C_P, expansion and isothermal modulus, and the mode-weighted Gamma contraction on a stable equilibrium family of the static-plus-harmonic model. Mode-Gruneisen producers, static-energy and phonon producers, numerical volume scans, EOS fits and global minimization remain unverified. The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "HOMOLUMOGap", "uid": "8998dd04e7ff8c51679a06f9c40bffb8923fc6d9b8b1f5b3b955999c029a5c5e", "binding_ids": [], "status": "gap", "reason": "No current Physlib molecular Hessian, electronic gap, NEB, or bond-dissociation formula identification; harmonic oscillator analogies remain conceptual." }, { "id": "HeatCapacity", "uid": "343adfdb0717f7f81ae52cc5fca8279ff84a9715ba19b5630d405453682b12f9", "binding_ids": [ "harmonic-mode-heat-capacity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "HeatCapacityConstantP", "uid": "08057e4554a3420c933329b3821de9295e2b5950191e23cafe310b0a15adc7b6", "binding_ids": [ "harmonic-qha-heat-capacity-p", "harmonic-static-cp-identity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "HeatCapacity[method=pimd]", "uid": "3064108b9fe3839c9ca154ba7887f73cd550c02944ddf61dc26fcbdf07cf6fe0", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "HeatCurrent", "uid": "aab7019fbc014398b74ece01428527aa95e89da6f6d51efc0323b0acf59248c6", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "HeatCurrentACF", "uid": "d8f44a4cce5e3dfea1ec9329189d49ac70e33d69cab222e2679f7791a30613ac", "binding_ids": [ "md-correlation-estimators" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "HelmholtzFreeEnergy", "uid": "38b082d42f5136a903752e5425c65c112c90bc564693ddb929ef3f175d9d96d8", "binding_ids": [ "harmonic-mode-free-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "InterfaceConductance", "uid": "2995dc3d963a6d387aa5b62d35a7248391ce3ce8047c0dc381cfc1cc99ef08f9", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified." }, { "id": "InterfaceConductance[transport_model=nemd]", "uid": "9dabd5c3e66ac0b87479050482b623c9d72de947eb2aa74730dcb7ad7b4c8d81", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified." }, { "id": "InterfaceTrapRelaxationTime", "uid": "d9e8d6ad94a3bd0629bc56fd2fbc5a6555e2866c3ff667aefd714ad75a27a491", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "InternalEnergy", "uid": "4e9c9eb989aa721167806f40b353dbb75aedba068a1a486eeef178f77432108d", "binding_ids": [ "harmonic-mode-internal-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "IsotopeAbundances", "uid": "f8e4253dc1f218daff8e76af5b433dc230bf952f910ba163f87c6fa331aa033d", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "Linewidth[channel=anharmonic_3ph]", "uid": "f52d161ad0be284c121c6fd10fe403c922aaa59646746c12719622004da5fa3a", "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Linewidth[channel=boundary]", "uid": "81288248a2b40f0abfe78b058ffa4025a0312324e8af26b801017b9a0971961e", "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Linewidth[channel=isotope]", "uid": "6cbd3322c5b6eb0105f9ca22d9ddc88334206ec9c115a31e6fb87409cc374c45", "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Linewidth[channel=total]", "uid": "50c33a1edd60bff6fb74124af1a8847a4c8a2c58d39554b4661e6a2cddb62f30", "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "MagneticMoment", "uid": "1423a9b16e18881e4388fee44f536891a6295aaa6e0fc3775bbbe89ee480847a", "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "MassDensity", "uid": "24749d8ecb8f22bd0931e83ebdc540a6cb0753bd4f3a611f142b31050353d42d", "binding_ids": [], "status": "gap", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved." }, { "id": "MeanFreeDisplacement[bte_solver=direct_inverse]", "uid": "b24a26725f5641ffb16720514a7fdb23bf03304ce0d51c9f39501d1c8c009b6c", "binding_ids": [ "linearized-bte-conductivity" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "MeanFreeDisplacement[bte_solver=rta]", "uid": "ad1373f31b9dbb8bd24ea16e1f90f1d92a39d309f2dc22583fad4fd82eca6f59", "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "MeanSquaredDisplacement", "uid": "28ebe96bc049a92e4318fd5242778f30e5647531d0a2b39c5891546329c8bca0", "binding_ids": [ "md-correlation-estimators" ], "status": "local", "reason": "No matching current Physlib Einstein-MSD, Arrhenius, Nernst-Einstein, or cluster-expansion formula theorem identified. The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "MigrationBarrier", "uid": "c59c8d2bc372392cff55803e7b8b742878dac82f3ab7623bd267acd006d23d07", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "ModalDiffusivity", "uid": "6052fe7e6aa8d2116b369886bffe59b04859f21145640184d24dfd96fbb60513", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "MolarEnthalpy", "uid": "600d0860290d3326202768db7249339c34ac675deb2ee8528a92207d3878b00c", "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "MolarEntropy", "uid": "0f08fcb37791dc7cc3e0e0ee5c611abbefc5075867719b275fcaa77ada4f1ac2", "binding_ids": [ "harmonic-joint-molar-ensemble", "harmonic-molar-entropy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "MolarGibbsEnergy", "uid": "b048bc7569d568e2201557761eb758817674dd3a38b6f2379c5474f4cf91c66f", "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "MolarHeatCapacity", "uid": "4fa2d8e16bde42f794f150f5d149e5d645d63f7ddaac59cef79bd41771b13c17", "binding_ids": [ "harmonic-joint-molar-ensemble", "harmonic-molar-heat-capacity", "harmonic-static-cp-identity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "MolarHelmholtzFreeEnergy", "uid": "419335bd5f3a1949576e608e4147d8f18bef980a69a9c6b99199f812c806b7be", "binding_ids": [ "harmonic-joint-molar-ensemble", "harmonic-molar-free-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "MolarInternalEnergy", "uid": "220926f4974baec5c511f5acae2c9c4e09a8276d41c1314391b91dec537ef2fb", "binding_ids": [ "harmonic-joint-molar-ensemble", "harmonic-molar-internal-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "MolarVolume", "uid": "e4ae49aad73336225c26fd3cf5d8dc2f4477b02b7d8bad22c9c74fd40a2482b2", "binding_ids": [ "harmonic-static-cp-identity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "MolecularFrequency", "uid": "e1270ea9a293c9107226bf831f60985f5525a0c386dca70841743bdc5d082c26", "binding_ids": [ "classical-harmonic-frequency", "molecular-normal-modes" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ParticipationRatio", "uid": "09138ae29cee41ae5f1a1758ac65a479625ef4d3c6026bf7ad5b60f884d668a0", "binding_ids": [ "participation-ratio-bounds" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "PhaseFraction", "uid": "7b55b11531084dcb2d5fd95100d62b2d68b787596e7c9712e06c011817ab5b8c", "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "PhaseSpace3Phonon", "uid": "ec483cecbdbfec9e13ce3b2033b03e4581edf7bc0ff935d09ab9c24a7e481080", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "PhononDOS", "uid": "e14b6cdd007ca3af971aa52880811138606efdb47e30d5a26c2335903ce796f3", "binding_ids": [ "crystal-dos-landauer-conductance", "vacf-spectrum-lorentzian-limit" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "PhononTransmission", "uid": "1e8b145f7ea4fc232aac842c985a46fee4a6d2b2a04118a38d67dce470ed1b50", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "PoissonRatio", "uid": "e95ba3f5810f8554656c148dae26520cf3258cace0b951dd75e69ee8c552c132", "binding_ids": [ "linear-elasticity-voigt-moduli" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Potential", "uid": "80a4273845e09207860e06db18d1f1ee5670a9b9c8b1020a99928c872878f901", "binding_ids": [ "classical-harmonic-potential-force" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "PotentialEnergyFitError", "uid": "d3f969394ab79173ea4afe796f00bc4b941141aa5f781e90fb3217e945f70755", "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "PotentialForceFitError", "uid": "833c3db3fbd145b45ed051cd6eaf782d7555518a0634f524b9ceee7fe49245e0", "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "PotentialOfMeanForce", "uid": "7a4fd6ca35f412c375006146204be07210a037901b0e4fb33e1805f243c74b6a", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "PotentialUncertaintyExceedance[method=committee_force_spread]", "uid": "e382fb86bf6c620d5af35954c8f189650ee0ca02dc3c389ed13cb533be8b76d9", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "PotentialVirialFitError", "uid": "8d85a09e82115bdd601be2ded9450446a9cdbdf0364bc88f421215019d00677a", "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "PowerFactor", "uid": "6c2382614be093f934d4f830419db9d60d32610041a8c213b677efc42f6a98ee", "binding_ids": [ "thermoelectric-power-factor" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "Pressure", "uid": "4ca6efdb874d146a6cca564c7a8a0504ee8868ae5890d4c33abddc147423d6f1", "binding_ids": [ "continuum-stress-sign", "thermodynamic-pressure" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ProgrammingPulseEnergy", "uid": "4dc358661b8a847db69d30a786b2b1475f1a291809bb34f55612d3203975be34", "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "QHAGibbsEnergy", "uid": "bb10912780eb22d45beff551be14f7f4216f3aae26161e8bfd60cdc051f63f58", "binding_ids": [ "harmonic-qha-gibbs-energy", "harmonic-qha-heat-capacity-p", "harmonic-qha-thermal-expansion" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "QuantumKineticEnergy", "uid": "a9a8eae4014c21afe87f7332c4aa842b0fd9002a7ed33e0bdb1ec339402fc9c9", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ReactionBarrier[construction=neb_mep]", "uid": "c922f2cfe105ec7d7fc5afd8eabcc0cc2e7adde25e2231272a4dd34493b4bd98", "binding_ids": [], "status": "gap", "reason": "No current Physlib molecular Hessian, electronic gap, NEB, or bond-dissociation formula identification; harmonic oscillator analogies remain conceptual." }, { "id": "ReactionEnergy", "uid": "a5736aa4760c27562f9e189829763a891b6fed83fe4a030a7e306b8d909d825c", "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "RetentionTime", "uid": "4280779257128d9b63d9180e7b7b03ab80beac66ee35480b178a09fa46b289f2", "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "SchottkyBarrierHeight[band_carrier=electron]", "uid": "a9e6a79e618d279fa3030c939d079eb130aa60268b1fc07a40393bec703496ca", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "SeebeckCoefficient", "uid": "80778c5d6cb9f0d1dd23d5c6cd6059acea161892532ce7c581c7477257debf02", "binding_ids": [ "boltzmann-electronic-transport", "thermoelectric-input-coefficients" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "SetVoltage", "uid": "5b46cb4fc1bf84f1fa97e2b52a352694f5c98c856e9b9034763243994a1c2d21", "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ShearModulus", "uid": "1ab85488c90143e38b94e27f93ab96571f381e7b09d507fa7a38216ffd34c5ec", "binding_ids": [ "linear-elasticity-voigt-moduli" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "StateCoefficientOfVariation", "uid": "165019489c9bfee532b8ff4c164d71f2a16516a933f3f40950554bdc8a0f84a5", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "StaticDielectricTensor", "uid": "41f6c971e70bc42117a79b8ec51039f498280fdd9b77ef433bb602f90f966d93", "binding_ids": [ "static-dielectric-ionic-response" ], "status": "local", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Stress", "uid": "183e66d4e443326d2c346a0327cdb125026bd8f6a91a9bb97b61265fcade7e85", "binding_ids": [ "cell-stress-rotation-balance", "continuum-stress-sign" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "Structure", "uid": "db1df71467f5a79440f849704ac20a33573ed4e270b5ebba59549272b2475239", "binding_ids": [ "abstract-crystal-structure" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "SurfaceEnergy", "uid": "306384aed4a1204fb7ca2347729de8d1816fa5fc32f4855a5d5f56a4d0fc1ddf", "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Temperature", "uid": "63f9bd32d94d77e12afbb96f5b1b7a870de6b1f5805213b7f3c3ff227da301f0", "binding_ids": [ "absolute-temperature", "harmonic-static-cp-identity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ThermalConductance[transport_model=landauer]", "uid": "50084cef8405999bb14c2ff71b98f70f982d71254e3ec35bed543a10bbbbf542", "binding_ids": [ "crystal-dos-landauer-conductance" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ThermalConductivity", "uid": "a03324d4def54619f16ee80ccf176b95a59808141651debabbf1611b7e3a12d5", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ThermalConductivity[bte_solver=direct_inverse]", "uid": "a369d33e9643b9f8766b111cde72c44fbf34a13ae46dfab31e1afab1c2129ca9", "binding_ids": [ "linearized-bte-conductivity", "thermoelectric-total-conductivity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ThermalConductivity[bte_solver=rta]", "uid": "d79d88d53266c147ea8d806cdbd30df309199469351ee13bcc9040a98745ebc4", "binding_ids": [ "canonical-rta-thermal-conductivity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ThermalConductivity[contribution=total]", "uid": "2931c0e41adb40b9ce80f5169881f3a266a2e85d5b280223914fb3b61628aea9", "binding_ids": [ "thermoelectric-total-conductivity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ThermalConductivity[effective_medium=nan,orientation=aligned]", "uid": "fc292985123b709c76547dc02721e60797cf5693fbad40b874826291314d5dd3", "binding_ids": [ "composite-effective-medium" ], "status": "local", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ThermalConductivity[effective_medium=nan,orientation=random]", "uid": "8cee68745e003909cd8e65688628af53a35c0803ace3f330394bbf0edf338d7b", "binding_ids": [ "composite-effective-medium" ], "status": "local", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ThermalConductivity[role=filler]", "uid": "1375fb6172571befa9a6551aa1d72bf541272a3f94ad6a762017ba3e4a44f78d", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified." }, { "id": "ThermalConductivity[role=matrix]", "uid": "b5cf0ca0f9b648341000b76e8839e45557973bb82a6e99460b05c066a448adf3", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified." }, { "id": "ThermalConductivity[transport_model=green_kubo]", "uid": "25dc9f7af8aec52efb6be16b7b7c76f8334b03056f8838c3ff432fdad8482fb6", "binding_ids": [ "md-correlation-estimators" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ThermalConductivity[transport_model=hnemd]", "uid": "e7157a52da5eb7d2ae79e57692acc30d7e63a3a0d1c731b14da4d6ab8fd5eb88", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ThermalConductivity[transport_model=nemd]", "uid": "d3e529fd2e7fa0a009440c21bd64949d6326c04eaa109fbed4cafac798608118", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ThermalConductivity[transport_model=qhgk]", "uid": "8932e3cbadef74a82802b43fd1d25328cb40e6136c90df3b6a4a998e77f9567b", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ThermalConductivity[transport_model=wigner]", "uid": "35e6000d8a0f2e4572e667befbd4cd5a8c6005fa8f6ef08791b6879c77baba64", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ThermalConductivity[transport_model=wigner_coherences]", "uid": "7f8d407ef3f36290c7f8e385cddb4a06a3c1bf34563f1df35ded2bb532e69fa4", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ThermalConductivity[transport_model=wigner_populations]", "uid": "3f837e061bf37d1ad104455abad3b9bd3d8bb16048a656977c2488fd3c4bd521", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ThermalExpansion", "uid": "9b42f8aebb6099cdcd910a272a92208881b598688bac7830c3ae0a412152473e", "binding_ids": [ "harmonic-qha-thermal-expansion", "harmonic-static-cp-identity", "harmonic-static-thermal-gruneisen-identity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "ThermalGruneisen", "uid": "22b1bd519d50d93fd24a919e021bab85aff246464a66a08f8274558834983985", "binding_ids": [ "harmonic-qha-thermal-gruneisen-contraction", "harmonic-static-thermal-gruneisen-identity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "TotalEnergy", "uid": "cf5a36298543de2027b173b757d38d80b38b66a85d2d3a56f55097e3da0e9a5e", "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source. No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. No current Physlib molecular Hessian, electronic gap, NEB, or bond-dissociation formula identification; harmonic oscillator analogies remain conceptual. The local quasiharmonic bridges cover the windowed Gibbs minimum, its C_P, expansion and isothermal modulus, and the mode-weighted Gamma contraction on a stable equilibrium family of the static-plus-harmonic model. Mode-Gruneisen producers, static-energy and phonon producers, numerical volume scans, EOS fits and global minimization remain unverified. No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified." }, { "id": "Trajectory", "uid": "12cde23e9a333fdd03313c4cdbd82d51033f9bb3bd505466a8db5360faf166b4", "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "TransitionTemperature", "uid": "8b9dd2a79db11c01e15d37f64a7bd0d9c95f09ea080a67f7912d9b8aabd7b132", "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "VelocityAutocorrelation", "uid": "f8b092359048e5130d56697ca9504bb4740b875aca685a8f760316bfb5e9a661", "binding_ids": [ "md-correlation-estimators" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "Voltage", "uid": "26831a0aabc3508a944405ec2fd763a0637abeeac1ed3d87df64c27392d638f4", "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "VolumetricHeatCapacity", "uid": "29e324b97f31cdb7d3cd8ef6aec5668fff111fd7ee96bd19dd0426b8728c1d47", "binding_ids": [ "harmonic-joint-volumetric-heat-capacity", "harmonic-static-thermal-gruneisen-identity", "harmonic-volumetric-heat-capacity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "WorkFunction", "uid": "bb5595d7a9f8f599458906c16f1b5ad255d70e7dc4a2001e54c626da2fc0ca70", "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "YoungsModulus", "uid": "022d18213f7d63713ef4fddb484e1ae6d898873ce20377176013f620225637d8", "binding_ids": [ "linear-elasticity-voigt-moduli" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "ZT", "uid": "cf5879d2e17024fb396d83e19e71852f6cedf446fcf9d2feb81eb994e3e66e3a", "binding_ids": [ "thermoelectric-figure-of-merit" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." } ], "operators": [ { "id": "apply_conductance_drift", "uid": "f1e70e363faaf1f396092f32ad11a3807bdc9e0a4a11c95e8479a9d069e4f2ec", "domain": "analog_device_metrology", "formula_sha256": "4fd6bd7d557005282f8b1babd293cc0d95e6fc85b93deeb6c49956a491a8718f", "formula_latex": "G_{c} = G_{0} \\left(\\frac{t_{d}}{t_{d,0}}\\right)^{- \\nu_{drift}}", "inputs": [ "ConductanceDriftExponent" ], "outputs": [ "ConductanceState" ], "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "apply_nac_correction", "uid": "b8fd10813fea32fe974f69f9bf3b2129e374e4556047828e7ffb768249e5bb12", "domain": "thermal_transport", "formula_sha256": "f81d4dc1d53be67dc0902b57c7bf91ec745a39a8d584827ffd023470de9e431f", "formula_latex": "{D}_{i,j,\\mathbf{q}} = {D^{bare}}_{i,j,\\mathbf{q}} + \\frac{4 \\pi q_{e}^{2} \\left(\\sum_{\\gamma=1}^{3} {Z^{*}}_{i,\\gamma,\\alpha} {q}_{\\gamma}\\right) \\sum_{\\delta=1}^{3} {Z^{*}}_{j,\\delta,\\beta} {q}_{\\delta}}{V_{cell} \\sqrt{{M}_{i} {M}_{j}} \\sum_{\\substack{1 \\leq \\gamma \\leq 3\\\\1 \\leq \\delta \\leq 3}} {\\varepsilon_{\\infty}}_{\\gamma,\\delta} {q}_{\\delta} {q}_{\\gamma}}", "inputs": [ "BareDynamicalMatrix", "BornCharges", "DielectricTensor" ], "outputs": [ "DynamicalMatrix" ], "binding_ids": [ "static-dielectric-ionic-response" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "autocorrelate_heat_current", "uid": "70b3a24785d80cd8cd1e3940c64032d39cad772ff92e73cd458f8b617ef40b1e", "domain": "thermal_transport", "formula_sha256": "44d85341be03b3ba6fccd6399e75dc62c09a61b36f4f3a2798f5d09b6d515a0e", "formula_latex": "{Jcorr}_{\\alpha,\\beta,\\tau} = \\frac{\\sum_{t=1}^{n_{lag}} {J}_{\\alpha,t} {J}_{\\beta,\\tau + t}}{n_{lag}}", "inputs": [ "HeatCurrent" ], "outputs": [ "HeatCurrentACF" ], "binding_ids": [ "md-correlation-estimators" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "combine_kappa_wigner", "uid": "d267d86e7f4e4a66d26cf3bab54f4d1a5a693d972562a85f2154e290ed5c8bf4", "domain": "thermal_transport", "formula_sha256": "b102eaf4d47db83cf47823ebc8a6bd8cfb28eb319b87238c461f326376906c07", "formula_latex": "{\\kappa^{W}}_{\\alpha,\\beta} = {\\kappa^{W,coh}}_{\\alpha,\\beta} + {\\kappa^{W,pop}}_{\\alpha,\\beta}", "inputs": [ "ThermalConductivity[transport_model=wigner_populations]", "ThermalConductivity[transport_model=wigner_coherences]" ], "outputs": [ "ThermalConductivity[transport_model=wigner]" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "compute_adsorption_energy", "uid": "35f53c511ef6202b79b33e89856071aa1f3f8610ad7705bd3f608784d6216a1f", "domain": "stability", "formula_sha256": "55f4d4ba2cc61605a32dbe38ea3b7883519a192efc4d173bca1a654bd09810f0", "formula_latex": "E_{ads} = E^{adslab}{\\left(E_{tot},\\mathcal{S} \\right)} - E^{adsorbate}{\\left(E_{tot},\\mathcal{S} \\right)} - E^{slab}_{ads}{\\left(E_{tot},\\mathcal{S} \\right)}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "AdsorptionEnergy" ], "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_band_gap", "uid": "25e07a60a3f641abc12987675165b1195ea8ed9d7adff7217736c7bdfd824b1a", "domain": "dft_ground_state", "formula_sha256": "a179ac399e42f7b16a3632a3a349f69da8990f135ae083d988faf5f076f22f64", "formula_latex": "E_{gap} = E^{gap}_{KS}{\\left(\\mathcal{S},V \\right)}", "inputs": [ "Structure", "Potential" ], "outputs": [ "BandGap" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "compute_bond_dissociation", "uid": "1795d2610a857e1ab6d3fff4c3fa60712330c92f6e370899bcf63813a417f805", "domain": "molecular", "formula_sha256": "8acbedb7959f2c012cd73222a0b1c61a5df4a733ada7043172d06e69bc8c888f", "formula_latex": "E_{BDE} = \\Delta_{BDE}{\\left(E_{tot},\\mathcal{S} \\right)}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "BondDissociationEnergy" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib molecular Hessian, electronic gap, NEB, or bond-dissociation formula identification; harmonic oscillator analogies remain conceptual." }, { "id": "compute_boundary_scattering", "uid": "1326da3355b506b5142329e3b074a09589b95e4e18d1819c4c44ad579c1fd41e", "domain": "thermal_transport", "formula_sha256": "4710be932ea5cc929dedfe2bb5cf1b7e240e1781fa12cc45f9dcdd33c6c8c78e", "formula_latex": "{\\Gamma^{bnd}}_{\\mathbf{q},\\nu} = \\frac{\\sqrt{\\sum_{\\alpha=1}^{3} {v}_{\\alpha,\\mathbf{q},\\nu}^{2}}}{2 L}", "inputs": [ "Frequency", "GroupVelocity" ], "outputs": [ "Linewidth[channel=boundary]" ], "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_bulk_modulus_eos", "uid": "e78d5769e8a4718966dee70a82eb5ac9622b35e20edf471ee1d8dc88ab40868c", "domain": "mechanics", "formula_sha256": "486f990eb13996a627d4274aada76d2ba5b3cdff7518587837b45ee121e12ace", "formula_latex": "K = V_{0} \\frac{d^{2}}{d V_{cell}^{2}} E_{tot}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "BulkModulus" ], "binding_ids": [ "linear-elasticity-voigt-moduli" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_bulk_modulus_qha", "uid": "de67212047d6c2bb08a94d9394adffe1b1cc183b7a1811d838de2dcfb9323235", "domain": "quasiharmonic", "formula_sha256": "bedbb8f82d002302668176dcdb9aa7abcbbe648e5963121237de436c725b2e7b", "formula_latex": "K = K^{qha}{\\left(E_{tot},\\mathcal{S},\\omega \\right)}", "inputs": [ "TotalEnergy", "Structure", "Frequency" ], "outputs": [ "BulkModulus" ], "binding_ids": [ "harmonic-qha-bulk-modulus" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_calphad_entropy", "uid": "c1a786957352c7a02efab90d3815c83156bbae53976932e51dec1a9f0df62f2b", "domain": "thermochemistry", "formula_sha256": "841219f58ee33183c89a9fcd83b3c4a34c426c7be1473d45431ccd81b756c7cf", "formula_latex": "S_{m} = S^{eq}{\\left(\\mathcal{D},T \\right)}", "inputs": [ "AssessedDatabase", "Temperature" ], "outputs": [ "CalphadMolarEntropy" ], "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_carrier_density", "uid": "3aa782c1767f3b70762cd54c7414e42f2e59193946d05c8ad8dcb565747a7600", "domain": "materials", "formula_sha256": "ff614aadedf48310e2bdebc32cf29ee95f26fe184f672cb6fbb2abe993c4f987", "formula_latex": "n_{c} = n^{c}{\\left(\\mathcal{S} \\right)}", "inputs": [ "Structure" ], "outputs": [ "CarrierDensity" ], "binding_ids": [], "status": "gap", "reason": "An external selector over the structure (which species are mobile, counted over the cell volume); nothing algebraic to prove beyond its dimension." }, { "id": "compute_carrier_mobility", "uid": "8de71440c5216dc390d9e5595d6e7dcc50a672fbc2c7c36400c04602cd1d2048", "domain": "electronic_transport", "formula_sha256": "21573bfd16d2a71a3fe127fb2685490564e562befd3119c79c02853ff268a121", "formula_latex": "\\mu_{e} = \\mu^{bte}{\\left(\\mathcal{S},\\varepsilon_{0},C,\\omega \\right)}", "inputs": [ "Structure", "StaticDielectricTensor", "ElasticConstants", "Frequency" ], "outputs": [ "CarrierMobility" ], "binding_ids": [ "electronic-carrier-statistics" ], "status": "local", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_chemical_potentials", "uid": "6e99a335cfb6b775c858856c9f81be42bf54bb931975aa16e4b13eedfa874b61", "domain": "thermochemistry", "formula_sha256": "a95d84ace50d2262f60cc554ca21e03163140ec272d406f065bd39d9d6610fcb", "formula_latex": "\\mu = \\mu^{eq}{\\left(G_{m} \\right)}", "inputs": [ "MolarGibbsEnergy" ], "outputs": [ "ChemicalPotential" ], "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_conductance[transport_model=landauer]", "uid": "1201b09100e9aaac69f343b09ff8c8002c6de6dec1c843458bec89e3151ef436", "domain": "thermal_transport", "formula_sha256": "ae627a4fd4b58e0fea7656eb1b7f74fb8e4377ca53f2de6f080e09ca05e57aef", "formula_latex": "\\mathcal{G} = \\frac{\\int\\limits_{0}^{\\infty} \\bar{\\h} \\omega \\frac{\\partial}{\\partial T} n_{BE}{\\left(\\frac{\\bar{\\h} \\omega}{T k_{B}} \\right)} {\\mathcal{T}}_{\\omega}\\, d\\omega}{2 \\pi}", "inputs": [ "PhononTransmission", "Frequency", "Temperature" ], "outputs": [ "ThermalConductance[transport_model=landauer]" ], "binding_ids": [ "crystal-dos-landauer-conductance" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_configurational_energy", "uid": "d3cc1efc62bc2eaeca95cbd9c0cc10a3f0bbdf850d7ef3c4991d437cdef4f084", "domain": "materials", "formula_sha256": "8cd14a60a26594aaa2e9b30a36bc13e9ed6f5a95cd65b2ffde43a259143ec1f9", "formula_latex": "E_{cfg} = E^{ce}{\\left(V,\\mathcal{S} \\right)}", "inputs": [ "Potential", "Structure" ], "outputs": [ "ConfigurationalEnergy" ], "binding_ids": [], "status": "gap", "reason": "An external cluster-expansion evaluation over a fitted checkpoint; nothing algebraic to prove beyond its dimension." }, { "id": "compute_dispersion", "uid": "1ab7a3aee9b25b786739afcd517b909e4fdf091c768ab4ecbea007e53633be92", "domain": "thermal_transport", "formula_sha256": "58532a304fabac6818bbb184abfe0a1f3b140583aec52ace8a1b4a73f1d8db5e", "formula_latex": "\\sum_{j=1}^{3 N} {D}_{i,j,\\mathbf{q}} {e}_{j,\\mathbf{q},\\nu} = {\\omega}_{\\mathbf{q},\\nu}^{2} {e}_{i,\\mathbf{q},\\nu}", "inputs": [ "DynamicalMatrix" ], "outputs": [ "Frequency", "Eigenvectors" ], "binding_ids": [ "crystal-phonon-dispersion" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_dos", "uid": "c42288d9daf6b41adc999cb7f35f8dc0d46478006f253df31c7aedf2605783bc", "domain": "thermal_transport", "formula_sha256": "8c3bd346a566aedaa4ea5dc0a812a8404cd1a6e7d221e4e7e5709550addeab01", "formula_latex": "g = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\delta{\\left(\\omega - {\\omega}_{\\mathbf{q},\\nu} \\right)}}{N_{q}}", "inputs": [ "Frequency" ], "outputs": [ "PhononDOS" ], "binding_ids": [ "crystal-dos-landauer-conductance" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_dynamical_matrix", "uid": "67d4d9c1ad01e2aa1705e54ed9edb2d1a0ce601aa0e7c9929a7ba7cbd7259f28", "domain": "thermal_transport", "formula_sha256": "d084d4d043a870f7a1d7dcb9931a150768819a4ce1f993c3620ad5b8e99e8897", "formula_latex": "{D^{bare}}_{i,j,\\mathbf{q}} = \\frac{\\sum_{\\mathbf{R}=-\\infty}^{\\infty} e^{i \\mathbf{R} \\mathbf{q}} {\\Phi^{(2)}}_{i,j,\\mathbf{R}}}{\\sqrt{{M}_{i} {M}_{j}}}", "inputs": [ "ForceConstants[order=2]" ], "outputs": [ "BareDynamicalMatrix" ], "binding_ids": [ "crystal-dynamical-matrix" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_elastic_constants", "uid": "3e995930ac921247acde0b3724b91c698f5c8e93c991635155d50a9c8039fe85", "domain": "mechanics", "formula_sha256": "266bfe9ac04cee4e494fb4002abaffafab63319c53a23218ce1196c566832d6c", "formula_latex": "{C}_{\\alpha,\\beta,\\gamma,\\delta} = - \\frac{\\partial}{\\partial {\\varepsilon^{str}}_{\\gamma,\\delta}} {\\sigma}_{\\alpha,\\beta}", "inputs": [ "Stress", "Structure" ], "outputs": [ "ElasticConstants" ], "binding_ids": [ "hyperelastic-elastic-constants" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_electronic_conductivity", "uid": "9b20bb39ed27457a9f0c5655ab1001b0abdcf86c4ab4883c87e8a54847e0ab72", "domain": "electronic_transport", "formula_sha256": "2de7ab9fcfdde2fac6c5e3ab0e88316483e11a22ae90687e08aa8897908580e0", "formula_latex": "\\sigma_{el} = \\sigma^{bte}{\\left(\\mathcal{S},\\varepsilon_{0},C,\\omega \\right)}", "inputs": [ "Structure", "StaticDielectricTensor", "ElasticConstants", "Frequency" ], "outputs": [ "ElectricalConductivity[carrier=electronic]" ], "binding_ids": [ "boltzmann-electronic-transport" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_electronic_dos", "uid": "ca35c6e4d21df105dfe83087ea98e03a5bad951006ada4fe722eea0a06a1cf7f", "domain": "electronic_transport", "formula_sha256": "b399aaa658591d443481b4cb89cc5eb171dee83c7744a60637547caacec53411", "formula_latex": "g_{E} = g_{el}{\\left(E,\\mathcal{S},V \\right)}", "inputs": [ "Structure", "Potential" ], "outputs": [ "ElectronicDOS" ], "binding_ids": [ "electronic-carrier-statistics" ], "status": "local", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_electronic_thermal_conductivity", "uid": "b24757dc6349ac2bcf8429ab264385f76d6895fc52baea4eb380f7d7e063ca11", "domain": "electronic_transport", "formula_sha256": "7ffb8ea9b51164875b7f3eaa724d017c8c846b3fd22f4da0befa6169596c9586", "formula_latex": "\\kappa_{e} = \\kappa_e^{bte}{\\left(\\mathcal{S},\\varepsilon_{0},C,\\omega \\right)}", "inputs": [ "Structure", "StaticDielectricTensor", "ElasticConstants", "Frequency" ], "outputs": [ "ElectronicThermalConductivity" ], "binding_ids": [ "boltzmann-electronic-transport" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_energy_above_hull", "uid": "4e7f15bf40f88fab20271ac052a22b491e7955ab3ebab9bb229713adf057a040", "domain": "stability", "formula_sha256": "8c6556dbca3a51a285840593e5ab35fa13c898956188a1beec9febac45ae722b", "formula_latex": "E_{hull} = d^{hull}{\\left(\\Delta H_{f},\\mathcal{S} \\right)}", "inputs": [ "FormationEnergy", "Structure" ], "outputs": [ "EnergyAboveHull" ], "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_entropy", "uid": "40d04377570142198d0a7f3d505de3b8fa6fdcf0418622be679aceec11b13264", "domain": "thermal_transport", "formula_sha256": "a1368172bfefa61b76a34da3b474fa48f11a728e01b851f0abf1961cd53e61c7", "formula_latex": "{s}_{\\mathbf{q},\\nu} = - k_{B} \\log{\\left(1 - e^{- \\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}}} \\right)} + \\frac{\\bar{\\h} n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}} \\right)} {\\omega}_{\\mathbf{q},\\nu}}{T}", "inputs": [ "Frequency", "Temperature" ], "outputs": [ "Entropy" ], "binding_ids": [ "harmonic-mode-entropy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_fc2_finite_displacement", "uid": "8c4e324d9fb31b0cde5f98b83effdc03f0fe77d04b9caf332ceecff6fc16ece1", "domain": "dft_ground_state", "formula_sha256": "ffda0ae6e61fea11db6294a25665260253cdb5d1af6e47e27ef07cd1bfba64e9", "formula_latex": "{\\Phi^{(2)}}_{i,j,\\mathbf{R}} = - \\frac{d}{d u_{i(0)}} F_{j(R)}", "inputs": [ "Forces", "Structure" ], "outputs": [ "ForceConstants[order=2]" ], "binding_ids": [ "finite-displacement-force-constants" ], "status": "local", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_force_constants[order=2]", "uid": "8ad53af4eefce437872dd08ee834962e6db76f7803ab50771b4985c0782c64d4", "domain": "thermal_transport", "formula_sha256": "2f30cf2ccf4f3b65e69e78385ddc551525d63f6f92ab6929cf5dfc387fc20dd9", "formula_latex": "{\\Phi^{(2)}}_{i,j,\\mathbf{R}} = \\frac{d^{2}}{d u_{j(R)}d u_{i(0)}} V{\\left(\\{u\\} \\right)}", "inputs": [ "Potential" ], "outputs": [ "ForceConstants[order=2]" ], "binding_ids": [ "force-constant-sum-rules" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_force_constants[order=3]", "uid": "979756d34934118442ec22671b68e036b5e9751c95ed110d3a508ba2dee95288", "domain": "thermal_transport", "formula_sha256": "aa562fd454620e74d58b4d9cdb0abd8076dc039e13714849f5af3d972d25e7f8", "formula_latex": "{\\Phi^{(3)}}_{i,j,k,\\mathbf{R},\\mathbf{R'}} = \\frac{d^{3}}{d u_{k(R')}d u_{j(R)}d u_{i(0)}} V{\\left(\\{u\\} \\right)}", "inputs": [ "Potential" ], "outputs": [ "ForceConstants[order=3]" ], "binding_ids": [ "force-constant-sum-rules" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_forces_hf", "uid": "59ae54527b2e735e5408e66eadb81c9f1fa29720ea39207564e388a44cd1b04c", "domain": "dft_ground_state", "formula_sha256": "198daf948e48b420f367655f4ab063d3da5f54dd475468ce249711afdef38d0b", "formula_latex": "{F^{at}}_{i,\\alpha} = - \\frac{d}{d {R^{at}}_{i,\\alpha}} E_{tot}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "Forces" ], "binding_ids": [ "hellmann-feynman-force-balance" ], "status": "local", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_formation_energy", "uid": "7cff61b1def5290a5fb1bcdfcbf984d8244627d0b33e996480dc9da24aba3637", "domain": "stability", "formula_sha256": "307463c9fb60cd1f05eda5e8738aaa7306207d750f1c168d5c88ef6dccfb3fa9", "formula_latex": "\\Delta H_{f} = E^{form}_{ref}{\\left(E_{tot},\\mathcal{S} \\right)}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "FormationEnergy" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified." }, { "id": "compute_free_energy", "uid": "b15ffc57501399eb9f1a0081d34ec1e16b579134c884d6dcf0b05434924e02eb", "domain": "thermal_transport", "formula_sha256": "919246645690d2a35e9330824756ff2cb9bae8fb4dfffa4b5f93b953a4ce178b", "formula_latex": "{f}_{\\mathbf{q},\\nu} = T k_{B} \\log{\\left(1 - e^{- \\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}}} \\right)} + \\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{2}", "inputs": [ "Frequency", "Temperature" ], "outputs": [ "HelmholtzFreeEnergy" ], "binding_ids": [ "harmonic-mode-free-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_grain_boundary_energy", "uid": "74ba9279dc65f3acd9e2be4ca13861b510212aa24a19aca4b29c65976003bf1b", "domain": "stability", "formula_sha256": "f9a65c5a348516a72f064663e00f68d094e9511ed620539de392d4377397aca6", "formula_latex": "\\gamma_{GB} = \\frac{- N_{GB} E^{bulk}_{GB}{\\left(E_{tot},\\mathcal{S} \\right)} + E^{GB}{\\left(E_{tot},\\mathcal{S} \\right)}}{2 A_{GB}}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "GrainBoundaryEnergy" ], "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_group_velocity", "uid": "3cf7b8aa48dca910c062ce17738ede9cb609bacd7bd685bae23ac698897e38c3", "domain": "thermal_transport", "formula_sha256": "0b3f5977eb1cc8e4d87d72b54acc6ad49ac808800d4c3fbe88b7e9408ab41887", "formula_latex": "{v}_{\\alpha,\\mathbf{q},\\nu} = \\frac{\\sum_{\\substack{1 \\leq j \\leq 3 N\\\\1 \\leq i \\leq 3 N}} \\overline{{e}_{i,\\mathbf{q},\\nu}} {\\partial D/\\partial q}_{i,j,\\alpha,\\mathbf{q}} {e}_{j,\\mathbf{q},\\nu}}{2 {\\omega}_{\\mathbf{q},\\nu}}", "inputs": [ "DynamicalMatrix", "Frequency", "Eigenvectors" ], "outputs": [ "GroupVelocity" ], "binding_ids": [ "crystal-group-velocity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_gruneisen", "uid": "990caec662f730a5b2a416aa96173e6d21c5fac35b8ad6b0f12d96ea614e4a49", "domain": "thermal_transport", "formula_sha256": "6944abfc2f59713a961ef1b31caf81bbe7a961bfeea3c9225d96c7d352e1b0c3", "formula_latex": "{\\gamma_{G}}_{\\mathbf{q},\\nu} = - \\frac{\\sum_{\\substack{1 \\leq i \\leq 3 N\\\\1 \\leq j \\leq 3 N\\\\1 \\leq k \\leq 3 N\\\\-\\infty \\leq \\mathbf{R} \\leq \\infty\\\\-\\infty \\leq \\mathbf{R'} \\leq \\infty}} \\frac{e^{i \\mathbf{R} \\mathbf{q}} \\overline{{e}_{i,\\mathbf{q},\\nu}} {\\Phi^{(3)}}_{i,j,k,\\mathbf{R},\\mathbf{R'}} {e}_{j,\\mathbf{q},\\nu} {r}_{k,\\mathbf{R'}}}{\\sqrt{{m}_{i} {m}_{j}}}}{6 {\\omega}_{\\mathbf{q},\\nu}^{2}}", "inputs": [ "ForceConstants[order=2]", "ForceConstants[order=3]", "Frequency", "Eigenvectors" ], "outputs": [ "Gruneisen" ], "binding_ids": [], "status": "gap", "reason": "Not covered. The Maradudin-Fein contraction gamma_{q nu} = -(1/(6 omega_{q nu}^2)) sum_{ijk} sum_{R R'} Phi3_{ijk}(R, R') e*_i e_j exp(i q.R) r_k(R')/sqrt(m_i m_j) is not shown to equal the derivative-defined gamma_i = -(V/omega_i) d omega_i/dV of the quasiharmonic bindings; FC3 and finite-difference producers evaluated at the reference geometry differ from gammas at the selected volume." }, { "id": "compute_heat_capacity", "uid": "3662d6183de400b2337d314499e31dd473b7b2ea3994290f21046e36bdcedfdb", "domain": "thermal_transport", "formula_sha256": "b28a7bc9aae3f2172b22319465791fdb88fcfc5f7c0b8c3fcdfac99a9ded8bd9", "formula_latex": "{c}_{\\mathbf{q},\\nu} = \\frac{\\bar{\\h}^{2} {\\omega}_{\\mathbf{q},\\nu}^{2}}{4 T^{2} k_{B} \\sinh^{2}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{2 T k_{B}} \\right)}}", "inputs": [ "Frequency", "Temperature" ], "outputs": [ "HeatCapacity" ], "binding_ids": [ "harmonic-mode-heat-capacity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_heat_capacity_p", "uid": "36100a86ef25aada0acd615151092323ba680b4a9791bb58d1175adcea96deb2", "domain": "quasiharmonic", "formula_sha256": "94c543b5191d7fa6737a2f2fcca9cc68bc4cc38530abd5c2d31b25b091bc4ccf", "formula_latex": "C_{P} = C_P^{qha}{\\left(G_{qha} \\right)}", "inputs": [ "QHAGibbsEnergy" ], "outputs": [ "HeatCapacityConstantP" ], "binding_ids": [ "harmonic-qha-heat-capacity-p" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_heat_current", "uid": "02678de87aa09f341bbac81cbc6b0504afe2c8f889e4eab5d0700c3fe09f7dd0", "domain": "thermal_transport", "formula_sha256": "067539295f171de91c320f410d5b4b33b98d0769cb3bcc66208ee58eba1ed17d", "formula_latex": "{J}_{\\alpha,t} = \\frac{\\sum_{i=1}^{N_{atoms}} {E}_{i,t} {v}_{i,\\alpha,t} + \\sum_{\\substack{1 \\leq i \\leq N_{atoms}\\\\1 \\leq \\beta \\leq 3}} {W}_{i,\\alpha,\\beta,t} {v}_{i,\\beta,t}}{V_{cell}}", "inputs": [ "Trajectory" ], "outputs": [ "HeatCurrent" ], "binding_ids": [], "status": "gap", "reason": "Not covered. The per-atom virial W_{i,alpha beta} ((1/2) sum_{j != i} r_{ij,alpha} F_{ij,beta} for a pair potential, with r_ij = r_i - r_j and F_ij the force on i from j; the form of Fan et al., Phys. Rev. B 92, 094301 (2015), for many-body potentials) and the Irving-Kirkwood, Hardy and virial decompositions of the energy current are not formalized; OMAIMDCorrelations proves nothing about this operator." }, { "id": "compute_homo_lumo_gap", "uid": "785759ece87a293204a322aa5c662467cc649f6a0b69b641542aba0fa6ddfff4", "domain": "molecular", "formula_sha256": "c7c199917aac2381c7a369b5f489bb53ab6e0eb1e178e58b807c420a7960f77a", "formula_latex": "E_{gap}^{mol} = \\Delta_{HL}{\\left(\\mathcal{S},V \\right)}", "inputs": [ "Structure", "Potential" ], "outputs": [ "HOMOLUMOGap" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib molecular Hessian, electronic gap, NEB, or bond-dissociation formula identification; harmonic oscillator analogies remain conceptual." }, { "id": "compute_intercalation_voltage", "uid": "a4bc99a6b2efc1df1400041ada33cfb59e97819f189e3501a99c06f9c5e5d49f", "domain": "stability", "formula_sha256": "7a3dd261cb0bf1bb623ff3a0c3a2a353a03573911235a56c05e01e021b6f4949", "formula_latex": "V_{avg} = \\frac{\\mu_{ion} n_{ion} + E^{empty}{\\left(E_{tot},\\mathcal{S} \\right)} - E^{full}{\\left(E_{tot},\\mathcal{S} \\right)}}{n_{ion} q_{e}}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "Voltage" ], "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_internal_energy", "uid": "ce45fb09bf9c3f8076a2064b90853e163f80b20a3f5c7506af913c560accbc53", "domain": "thermal_transport", "formula_sha256": "6f0d545bd77043c0fa3b62ff5f97422a626b5ddd71b88baffe4d562b94d8f280", "formula_latex": "{e}_{\\mathbf{q},\\nu} = \\bar{\\h} \\left(n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q},\\nu}}{T k_{B}} \\right)} + \\frac{1}{2}\\right) {\\omega}_{\\mathbf{q},\\nu}", "inputs": [ "Frequency", "Temperature" ], "outputs": [ "InternalEnergy" ], "binding_ids": [ "harmonic-mode-internal-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_ionic_conductivity", "uid": "d4303ca638ffb5ca529f8c681909d728647149d1b79fd02ec8a18c9270afc05c", "domain": "materials", "formula_sha256": "8beec3f9dffff073f2679da513e2faea9a31883e2e816ad3815c04d18cdc6af5", "formula_latex": "\\sigma_{ion} = \\frac{D e_{c}^{2} n_{c} z^{2}}{T k_{B}}", "inputs": [ "CarrierDensity", "Diffusivity", "Temperature" ], "outputs": [ "ElectricalConductivity[carrier=ionic]" ], "binding_ids": [ "arrhenius-ionic-transport" ], "status": "local", "reason": "No matching current Physlib Einstein-MSD, Arrhenius, Nernst-Einstein, or cluster-expansion formula theorem identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_isotope_scattering", "uid": "7cf53f521738578b03e17d568ddf5db3df454b0a47bd4d621390c5c9250dec73", "domain": "thermal_transport", "formula_sha256": "bbcc0cd3146146e003b4319f36af2916f769fadd9449a39309e565afc2b97b21", "formula_latex": "{\\Gamma^{iso}}_{\\mathbf{q},\\nu} = \\frac{\\pi {\\omega}_{\\mathbf{q},\\nu}^{2} \\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N}} \\delta{\\left(- {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)} \\sum_{b=1}^{N} \\left(\\left|{\\sum_{\\alpha=1}^{3} \\overline{{e}_{\\alpha + 3 b - 3,\\mathbf{q},\\nu}} {e}_{\\alpha + 3 b - 3,\\mathbf{q'},\\nu'}}\\right|^{2}\\right) {g}_{b}}{4 N_{q}}", "inputs": [ "Frequency", "Eigenvectors", "IsotopeAbundances" ], "outputs": [ "Linewidth[channel=isotope]" ], "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_kappa[transport_model=qhgk]", "uid": "d610d8d258da853680448fe2c9507736608d2cae92578e31b079451060e6d528", "domain": "thermal_transport", "formula_sha256": "e6430c03f72538434979aea284edc0a3fc27d02adf584d25046899c6bdb59f53", "formula_latex": "{\\kappa^{QHGK}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N\\\\1 \\leq \\nu' \\leq 3 N}} \\frac{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right) {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu} {v}_{\\beta,\\mathbf{q},\\nu'}}{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right)^{2} + \\left(- {\\omega}_{\\mathbf{q},\\nu'} + {\\omega}_{\\mathbf{q},\\nu}\\right)^{2}}}{N_{q} V_{cell}}", "inputs": [ "HeatCapacity", "Frequency", "GroupVelocity", "Linewidth[channel=total]", "Temperature" ], "outputs": [ "ThermalConductivity[transport_model=qhgk]" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "compute_kappa[transport_model=wigner_coherences]", "uid": "fd0c5bbfc433db7ed55562ae46c06b3880389135fa84225a10c0687269251352", "domain": "thermal_transport", "formula_sha256": "a3e5a0b4ca99dddf01ec70ae4baab6a0583ce6945dae09681e50776d169c19a1", "formula_latex": "{\\kappa^{W,coh}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N\\\\1 \\leq \\nu' \\leq 3 N}} \\frac{\\left(\\frac{{c}_{\\mathbf{q},\\nu}}{{\\omega}_{\\mathbf{q},\\nu}} + \\frac{{c}_{\\mathbf{q},\\nu'}}{{\\omega}_{\\mathbf{q},\\nu'}}\\right) \\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right) \\left(\\frac{{\\omega}_{\\mathbf{q},\\nu'}}{2} + \\frac{{\\omega}_{\\mathbf{q},\\nu}}{2}\\right) {v}_{\\alpha,\\mathbf{q},\\nu} {v}_{\\beta,\\mathbf{q},\\nu'}}{\\left({\\Gamma^{anh}}_{\\mathbf{q},\\nu'} + {\\Gamma^{anh}}_{\\mathbf{q},\\nu}\\right)^{2} + \\left(- {\\omega}_{\\mathbf{q},\\nu'} + {\\omega}_{\\mathbf{q},\\nu}\\right)^{2}}}{N_{q} V_{cell}}", "inputs": [ "HeatCapacity", "Frequency", "GroupVelocity", "Linewidth[channel=total]" ], "outputs": [ "ThermalConductivity[transport_model=wigner_coherences]" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "compute_kappa[transport_model=wigner_populations]", "uid": "651c2374a726dda508bee50396ad83e8c635f98f73012f18397b319710d22846", "domain": "thermal_transport", "formula_sha256": "d10fe51fb540b288230d0aec858344b99e9d3c9d9f26af41836e435945f1aa8b", "formula_latex": "{\\kappa^{W,pop}}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "inputs": [ "HeatCapacity", "GroupVelocity", "MeanFreeDisplacement[bte_solver=direct_inverse]" ], "outputs": [ "ThermalConductivity[transport_model=wigner_populations]" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "compute_linewidth[channel=anharmonic_3ph]", "uid": "bed23822fc14b9f811e9d35b8361d56e0e38a34327b9e5b2c4e7f5aad6eebb72", "domain": "thermal_transport", "formula_sha256": "c5b1c700e285f83bade4999b264e709ae736be2f329f4eabb6ab45a86980b7e4", "formula_latex": "{\\Gamma}_{\\mathbf{q},\\nu} = \\frac{\\pi \\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N\\\\1 \\leq \\nu'' \\leq 3 N}} \\left(\\left(- 2 n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''}}{T k_{B}} \\right)} + 2 n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q'},\\nu'}}{T k_{B}} \\right)}\\right) \\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} + {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)} + \\left(n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''}}{T k_{B}} \\right)} + n_{BE}{\\left(\\frac{\\bar{\\h} {\\omega}_{\\mathbf{q'},\\nu'}}{T k_{B}} \\right)} + 1\\right) \\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} - {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)}\\right) \\operatorname{|V}^{2}_{3|}{\\left(\\mathbf{q},\\nu,\\mathbf{q'},\\nu',- \\mathbf{q'} + \\mathbf{q},\\nu'' \\right)}}{2 N_{q} \\bar{\\h}^{2}}", "inputs": [ "Frequency", "Eigenvectors", "ForceConstants[order=3]", "Temperature" ], "outputs": [ "Linewidth[channel=anharmonic_3ph]" ], "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_magnetic_moments", "uid": "06031f46aa761681e4ae480682ceeda163e0cd23aa69453db6ab01bf3ec3e4e6", "domain": "dft_ground_state", "formula_sha256": "0d78fde19e429350d4e5ff071fd67cd404d5e4d792877df2616d99256a94f048", "formula_latex": "{m^{spin}}_{i} = m^{KS}{\\left(\\mathcal{S},V \\right)}", "inputs": [ "Structure", "Potential" ], "outputs": [ "MagneticMoment" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "compute_modal_diffusivity", "uid": "decf7f445e4993527dcca3c243b35b3f328eaf16cac38d1668caf959e9f03975", "domain": "thermal_transport", "formula_sha256": "3fade1262c61e431ce4635fcc51f4146c96980d8d60bad622cc4c0b907f18443", "formula_latex": "D_{mode} = D^{QHGK}{\\left(\\omega,e,\\Gamma^{tot} \\right)}", "inputs": [ "Frequency", "Eigenvectors", "Linewidth[channel=total]" ], "outputs": [ "ModalDiffusivity" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "compute_molar_enthalpy", "uid": "901873fa2497cc75e75079357e7125527526bf2590507835f5cbeae3e5fd2868", "domain": "thermochemistry", "formula_sha256": "559358c8898996d1b25239cf7891b8cad1297e6964a661b04055f8cf53e78812", "formula_latex": "H_{m} = H^{eq}{\\left(\\mathcal{D},T \\right)}", "inputs": [ "AssessedDatabase", "Temperature" ], "outputs": [ "MolarEnthalpy" ], "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_molecular_frequencies", "uid": "e8e86bed52c1e09311b9baada0501b7668b72cf8de3797735bc4c7bb60090f3f", "domain": "molecular", "formula_sha256": "1b2f3949e74ab2b2db0136deee0dd84f2f576b76d18134a7624ebba18964fce0", "formula_latex": "{\\nu_{mol}}_{m} = \\nu^{H}{\\left(\\mathcal{S},V \\right)}", "inputs": [ "Structure", "Potential" ], "outputs": [ "MolecularFrequency" ], "binding_ids": [ "molecular-normal-modes" ], "status": "local", "reason": "No current Physlib molecular Hessian, electronic gap, NEB, or bond-dissociation formula identification; harmonic oscillator analogies remain conceptual. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_msd", "uid": "ce9b639e5b88ed7e27cd2e0c879c620c582b0435d01365ea1dc35eb3fafdeeaa", "domain": "thermal_transport", "formula_sha256": "b4f9b442a0af59ae09e92327dd9753d2a819fddaa612827d64c5dfb7d03a7146", "formula_latex": "{M}_{\\tau} = \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq i \\leq N_{atoms}\\\\1 \\leq t \\leq n_{lag}}} \\left({r}_{i,\\alpha,\\tau + t} - {r}_{i,\\alpha,t}\\right)^{2}}{N_{atoms} n_{lag}}", "inputs": [ "Trajectory" ], "outputs": [ "MeanSquaredDisplacement" ], "binding_ids": [ "md-correlation-estimators" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_participation_ratio", "uid": "93c6afe349f877656a16d18074e195ac4257226ce9d863d7f5d10ce96b679715", "domain": "thermal_transport", "formula_sha256": "5f3e67dbd9381917a0ab1c443fbf2698a741f2b3ce881210caac47777a0d07a9", "formula_latex": "{p}_{\\mathbf{q},\\nu} = \\frac{1}{N \\sum_{i=1}^{N} {a}_{i,\\mathbf{q},\\nu}^{2}}", "inputs": [ "Eigenvectors" ], "outputs": [ "ParticipationRatio" ], "binding_ids": [ "participation-ratio-bounds" ], "status": "local", "reason": "No Physlib model covers this formula. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_phase_fractions", "uid": "d5a6a4d1c789d427aecbc40998ec2b7d32240c2850ac5d9ed226e268b6ed09f8", "domain": "thermochemistry", "formula_sha256": "cdec0b486b6a7f7bbcc6f823a398f698ab01889104b0374971dd06ab7bf1397a", "formula_latex": "NP = f^{eq}{\\left(\\mathcal{D},T \\right)}", "inputs": [ "AssessedDatabase", "Temperature" ], "outputs": [ "PhaseFraction" ], "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_phase_space_3phonon", "uid": "cff24024635e104959182d2de98186efbc8cd8bf678ea239b1cc5ab24e76ff20", "domain": "thermal_transport", "formula_sha256": "9bcab6f3f298e5b0ee061cc1f7f751b1f4ff8634e63a5911b84feda66c0ab9ba", "formula_latex": "{P_{3}}_{\\mathbf{q},\\nu} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N\\\\1 \\leq \\nu'' \\leq 3 N}} \\left(\\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} - {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)} + 2 \\delta{\\left(- {\\omega}_{- \\mathbf{q'} + \\mathbf{q},\\nu''} + {\\omega}_{\\mathbf{q'},\\nu'} + {\\omega}_{\\mathbf{q},\\nu} \\right)}\\right)}{N_{q}}", "inputs": [ "Frequency" ], "outputs": [ "PhaseSpace3Phonon" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "compute_potential_energy_fit_error", "uid": "259d51fed7bf820b6edf9878f31f419e8921e05334f10c22e85a2bacc55ee514", "domain": "dft_ground_state", "formula_sha256": "b23a5b8a32a88e9c464b3eb0fd705740663f8a919c10349bf30c12b9d0ab45d9", "formula_latex": "\\epsilon_{E} = \\epsilon^{rmse}{\\left(V,E_{tot},N \\right)}", "inputs": [ "Potential", "TotalEnergy" ], "outputs": [ "PotentialEnergyFitError" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "compute_potential_force_fit_error", "uid": "617fa87d7443a62e99db43b88cbad44bb7179b37481c2711969dacbf744cf712", "domain": "dft_ground_state", "formula_sha256": "a368daf061e8a57f21222dd4965ae63944646097e55624c5a307db839cd8aaf7", "formula_latex": "\\epsilon_{F} = \\epsilon^{rmse}{\\left(V,{F^{at}}_{i,\\alpha} \\right)}", "inputs": [ "Potential", "Forces" ], "outputs": [ "PotentialForceFitError" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "compute_potential_uncertainty_exceedance", "uid": "cdd7b5cc99b5cc279c21df1c1e6b179a7eafb368b4b5f1d154fbd30d64fa8aef", "domain": "thermal_transport", "formula_sha256": "3fc1370058cde7dbf382dc689578197f25929f3ee52bd6a87f37f9cb5dc59a38", "formula_latex": "f_{\\sigma} = f^{exc}{\\left(V{\\left(\\{u\\} \\right)},r \\right)}", "inputs": [ "Potential", "Trajectory" ], "outputs": [ "PotentialUncertaintyExceedance[method=committee_force_spread]" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "compute_potential_virial_fit_error", "uid": "37e91b493c5bf8ebdb8c33f69f86fd14175ec025d32a8f087e3cdaa052809ad4", "domain": "dft_ground_state", "formula_sha256": "47e37f77222109f71c414a0549041be7cc26b6d398964dc82de22b8492331118", "formula_latex": "\\epsilon_{W} = \\epsilon^{rmse}{\\left(V,{\\sigma}_{\\alpha,\\beta},V_{cell},N \\right)}", "inputs": [ "Potential", "Stress" ], "outputs": [ "PotentialVirialFitError" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "compute_qha_gibbs", "uid": "2dfd344c598d659dbc5e4c8def4cefcb69b1a0bd215a9202f981618b18611b89", "domain": "quasiharmonic", "formula_sha256": "8264d933a1f5cf6dab090041c2a1e6367cd92e5de6022aaa8e7dbbbe20ed840e", "formula_latex": "G_{qha} = G^{qha}{\\left(E_{tot},\\mathcal{S},\\omega \\right)}", "inputs": [ "TotalEnergy", "Structure", "Frequency" ], "outputs": [ "QHAGibbsEnergy" ], "binding_ids": [ "harmonic-qha-gibbs-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_reaction_barrier", "uid": "15a6a24bb6b4ad6d1c1b44ef4d56407bd37565a4fa21a97efb99b763d75f3771", "domain": "molecular", "formula_sha256": "8099ab1fdbdc31cbbd0e87ac3bc1cd6c58829afb3a92b4d9eac2758506cc053d", "formula_latex": "E_{barrier} = \\Delta_{NEB}{\\left(E_{tot},\\mathcal{S} \\right)}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "ReactionBarrier[construction=neb_mep]" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib molecular Hessian, electronic gap, NEB, or bond-dissociation formula identification; harmonic oscillator analogies remain conceptual." }, { "id": "compute_reaction_energy", "uid": "2204ada808d0fae7706fe31723755bcc9614f2901bff9230e163f47a7e95cdf7", "domain": "stability", "formula_sha256": "c69c9c84d6d962cea6ea1f005df0612832d065d6fa4d759dfff5c9b1e08cbe1f", "formula_latex": "\\Delta E_{rxn} = \\sum_{i=1}^{N_{rxn}} {N^{fu}}_{i} {\\Delta H_{f}}_{i} {c_{rxn}}_{i}", "inputs": [ "FormationEnergy" ], "outputs": [ "ReactionEnergy" ], "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_seebeck", "uid": "f34ee134bfed051ae79dfece5f9ff45df4c2c202e784ccf6c3bf50035709d208", "domain": "electronic_transport", "formula_sha256": "75307432c842b2c58ffc978745dd0992e89936a44fab7dc40bb06a24a887b939", "formula_latex": "S = S^{bte}{\\left(\\mathcal{S},\\varepsilon_{0},C,\\omega \\right)}", "inputs": [ "Structure", "StaticDielectricTensor", "ElasticConstants", "Frequency" ], "outputs": [ "SeebeckCoefficient" ], "binding_ids": [ "boltzmann-electronic-transport" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_static_dielectric", "uid": "f658f73a798370ff39c363aef9b0d2219dd68d76251a561ee42d3b41e2d84a4a", "domain": "electronic_transport", "formula_sha256": "1aeb264ede74c9b48a916c554fbae4dfba2344232049591cd7231a254307fddd", "formula_latex": "\\varepsilon_{0} = \\varepsilon_0^{stat}{\\left(\\varepsilon_{\\infty},Z^{*},\\omega,e \\right)}", "inputs": [ "DielectricTensor", "BornCharges", "Frequency", "Eigenvectors" ], "outputs": [ "StaticDielectricTensor" ], "binding_ids": [ "static-dielectric-ionic-response" ], "status": "local", "reason": "Physlib has no electronic scattering, density-of-states or dielectric model; the Boltzmann coefficients are anchored to Physlib's ThermoelectricMaterial for the coefficient axioms and the figure of merit, and the mobility, electronic DOS and static dielectric tensor carry local proofs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_stress_cell", "uid": "88f8474bfda3d73e220f1db0ddfb8f00546007f1172ec663f4d4406568b3c5f2", "domain": "dft_ground_state", "formula_sha256": "6dd196c0b221e9aed1dae7019b053498c5d5d96177025dc088250bca9555842f", "formula_latex": "{\\sigma}_{\\alpha,\\beta} = - \\frac{\\frac{d}{d {\\varepsilon^{str}}_{\\alpha,\\beta}} E_{tot}}{V_{cell}}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "Stress" ], "binding_ids": [ "cell-stress-rotation-balance" ], "status": "local", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_surface_energy", "uid": "c8b2e006408e3e2c08f6c71eb7cd4e1a07f72ab20034fe883c691c1cad278492", "domain": "stability", "formula_sha256": "c99d3c4dab85511ea00104a21376fad96e0f80ce1982fddad3ac9f531454e8eb", "formula_latex": "\\gamma_{surf} = \\frac{- N_{slab} E^{bulk}{\\left(E_{tot},\\mathcal{S} \\right)} + E^{slab}{\\left(E_{tot},\\mathcal{S} \\right)}}{2 A_{surf}}", "inputs": [ "TotalEnergy", "Structure" ], "outputs": [ "SurfaceEnergy" ], "binding_ids": [ "dft-stability-energy-differences" ], "status": "local", "reason": "No current Physlib matching reference-state, convex-hull, surface, adsorption, reaction, or grain-boundary formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_thermal_expansion", "uid": "4e11a6ef1474661ab6ea320a5c99e8774edbe32b5a4dcdb13d97fda4b108995e", "domain": "quasiharmonic", "formula_sha256": "7a5492ca3cc69da193307f5f74b36985627e5b4ab5c97bc2851120f491553f93", "formula_latex": "\\alpha_{V} = \\alpha^{qha}{\\left(G_{qha} \\right)}", "inputs": [ "QHAGibbsEnergy" ], "outputs": [ "ThermalExpansion" ], "binding_ids": [ "harmonic-qha-thermal-expansion" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "compute_transition_temperature", "uid": "6f9aee9e741ba70cb9ee59098485a719edfaa20a864e0a5122ae7a225a875363", "domain": "thermochemistry", "formula_sha256": "dff18b12ad9924ead14269682911a8017a719684f240433725bdcdf9b4789c80", "formula_latex": "T_{\\mathrm{trans}} = T^{trans}{\\left(NP,T \\right)}", "inputs": [ "PhaseFraction", "Temperature" ], "outputs": [ "TransitionTemperature" ], "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database correspondence identified. The local proof covers only a unary system or a congruent transformation at one composition, the equal-Gibbs-energy temperature; the alloy boundary locus of the catalog (liquidus, solidus, solvus, where a phase fraction reaches 0 or 1) is not proved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "compute_velocity_autocorrelation", "uid": "9c182752f0970df148341ff0e66df8e5be8c74744e79d9626847399355dab7a2", "domain": "thermal_transport", "formula_sha256": "f3c1b5e43f2f5fc7b9ee7b7b2f2202e3b9b282fec518c74a87f993d3ddb3c758", "formula_latex": "{Cv}_{\\tau} = \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq i \\leq N_{atoms}\\\\1 \\leq t \\leq n_{lag}}} {v}_{i,\\alpha,\\tau + t} {v}_{i,\\alpha,t}}{N_{atoms} n_{lag}}", "inputs": [ "Trajectory" ], "outputs": [ "VelocityAutocorrelation" ], "binding_ids": [ "md-correlation-estimators" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "contract_bulk_modulus", "uid": "646fa795ffeed1fc8935994dd17159062c596fc0952ebb2a3159b3a1a74946fa", "domain": "mechanics", "formula_sha256": "461564a49eea3bd54eae3039dc5c24685f31ed2fbd89c5306cafadf7c74d2122", "formula_latex": "K = \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq \\beta \\leq 3}} {C}_{\\alpha,\\alpha,\\beta,\\beta}}{9}", "inputs": [ "ElasticConstants" ], "outputs": [ "BulkModulus" ], "binding_ids": [ "linear-elasticity-voigt-moduli" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "contract_conductance_window", "uid": "99e6e0bd3c2d868c2c7ce9ebad11cca90df15a0c005c7a362518c7d18ed0bada", "domain": "analog_device_metrology", "formula_sha256": "ef6f24de0d38486e9ca19a9becddb16457f75bedb55de8ea1b7a574c7b05eaca", "formula_latex": "W_{G} = \\frac{G_{c}}{G_{min}}", "inputs": [ "ConductanceState" ], "outputs": [ "ConductanceWindow" ], "binding_ids": [], "status": "gap", "reason": "A closed-form definition with nothing non-obvious to prove; no Lean theorem restates it, and it is not bound." }, { "id": "contract_cumulative_kappa[wrt=mfp]", "uid": "ccd6d5886ed78dc12e752ceb9d573aba2e79cdb99b147ee186abdbf16f4d11a2", "domain": "thermal_transport", "formula_sha256": "b049fcd2d918152aeb5bb6316f7d16a97aa8ac7dd9c5b6d14b00d95a81bda27f", "formula_latex": "{\\kappa^{cum}_\\Lambda}_{\\alpha,\\beta,\\Lambda_{c}} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\left(\\theta{\\left(\\Lambda_{c} - \\sqrt{\\sum_{\\alpha=1}^{3} {F}_{\\alpha,\\mathbf{q},\\nu}^{2}} \\right)}\\right) {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "inputs": [ "HeatCapacity", "GroupVelocity", "MeanFreeDisplacement[bte_solver=direct_inverse]" ], "outputs": [ "CumulativeKappa[wrt=mfp]" ], "binding_ids": [], "status": "gap", "reason": "Not covered. The cumulative operators take the linearized-BTE F (MeanFreeDisplacement[bte_solver=direct_inverse]), for which kappa_cum is a signed partial sum, so it is a distribution only when every modal contribution is nonnegative, as in the relaxation-time approximation; the descriptions now say so. OMAIPhyslibKappaContraction proves an exact three-mode linearized BTE (one process 1+2 <-> 3, Casimir boundary scattering, Bose-Einstein weights, symmetric positive-definite collision matrix) whose cumulative kappa_xx is not monotone in the cutoff, in both the frequency and the mean-free-path binning." }, { "id": "contract_cumulative_kappa[wrt=omega]", "uid": "0547586f6514cf3d00afe174e20fb0222f1a645a124957664f7de6dd7b232941", "domain": "thermal_transport", "formula_sha256": "b98fe42af9ace1d64fe087a9b7d20ba07b6bb458c62f7056a47f7b7578ad191e", "formula_latex": "{\\kappa^{cum}_\\omega}_{\\alpha,\\beta,\\omega_{c}} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} \\theta{\\left(\\omega_{c} - {\\omega}_{\\mathbf{q},\\nu} \\right)} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "inputs": [ "HeatCapacity", "Frequency", "GroupVelocity", "MeanFreeDisplacement[bte_solver=direct_inverse]" ], "outputs": [ "CumulativeKappa[wrt=omega]" ], "binding_ids": [], "status": "gap", "reason": "Not covered. The cumulative operators take the linearized-BTE F (MeanFreeDisplacement[bte_solver=direct_inverse]), for which kappa_cum is a signed partial sum, so it is a distribution only when every modal contribution is nonnegative, as in the relaxation-time approximation; the descriptions now say so. OMAIPhyslibKappaContraction proves an exact three-mode linearized BTE (one process 1+2 <-> 3, Casimir boundary scattering, Bose-Einstein weights, symmetric positive-definite collision matrix) whose cumulative kappa_xx is not monotone in the cutoff, in both the frequency and the mean-free-path binning." }, { "id": "contract_density", "uid": "a611c687b4bef634ea1f629f5dc1a4746e4abb8489d1ead4933714a331fff141", "domain": "mechanics", "formula_sha256": "e60bf0d8066fbee6e81ae145054146c412a37cf4c4034947557865f28db9814c", "formula_latex": "\\rho = \\rho^{cell}{\\left(\\mathcal{S} \\right)}", "inputs": [ "Structure" ], "outputs": [ "MassDensity" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved." }, { "id": "contract_device_conductance", "uid": "7f5e524af6d302713934d4a2bd5901abc64efdad9a5f8822bd19fe28b5b27cdf", "domain": "analog_device_metrology", "formula_sha256": "a57aff2c7d45172298a807999f389328df1fbd97817154a86ab9c138a420dd19", "formula_latex": "G_{c} = \\frac{A_{g} \\sigma_{el}}{L_{g}}", "inputs": [ "ElectricalConductivity[carrier=electronic]" ], "outputs": [ "ConductanceState" ], "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "contract_diffusivity", "uid": "0888c8ca7d7127d39bd7181a531fb1e773221ff5eba025c309772952ba5f4f16", "domain": "materials", "formula_sha256": "4dd53c755289559068e7ea6ddbe62ad86904968174b9b7855d8023f416473705", "formula_latex": "D = \\frac{\\mathrm{slope}_{MSD}}{2 d}", "inputs": [ "MeanSquaredDisplacement" ], "outputs": [ "Diffusivity" ], "binding_ids": [], "status": "gap", "reason": "A closed-form definition with nothing non-obvious to prove; no Lean theorem restates it, and it is not bound." }, { "id": "contract_gibbs_hts", "uid": "bb3f3ca8c5f842995d1e84cb06a7ebfc7073b0963abff93daba9352b5fe505ec", "domain": "thermochemistry", "formula_sha256": "49db2b3924555a40d1e2dcb7040d436316d89446195ea9ef8173638bfb403290", "formula_latex": "G_{m} = H_{m} - S_{m} T", "inputs": [ "MolarEnthalpy", "Temperature", "CalphadMolarEntropy" ], "outputs": [ "MolarGibbsEnergy" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs." }, { "id": "contract_heat_capacity_p_identity", "uid": "79a11e58595c8b262c104330dcfc622f13a787da6bae856cf4f40b57a6a93639", "domain": "thermodynamic_identities", "formula_sha256": "97ed2e0d3a11efad01a3ab0017d21be3da3009446b0b9364673fad4b8fdcb0f1", "formula_latex": "C_{P} = C_V^{mol} + K T V_{m} \\alpha_{V}^{2}", "inputs": [ "MolarHeatCapacity", "Temperature", "MolarVolume", "ThermalExpansion", "BulkModulus" ], "outputs": [ "HeatCapacityConstantP" ], "binding_ids": [ "harmonic-static-cp-identity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_interface_conductance[transport_model=nemd]", "uid": "46112f04a2831058c99b74c42a6daa0484ff1bcd14fce01c08deec4b05ffe73b", "domain": "composites", "formula_sha256": "b5afe85da9a62d7ffde89d0be4d0d09c149f24446ccc99b7cf4388d4d4006fda", "formula_latex": "G_{int} = \\frac{q_{\\mathrm{int}}}{\\Delta T_{\\mathrm{int}}}", "inputs": [ "HeatCurrent", "Temperature" ], "outputs": [ "InterfaceConductance[transport_model=nemd]" ], "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified." }, { "id": "contract_kappa[bte_solver=direct_inverse]", "uid": "8bd0005f31f01cd5d358f7cc58b21f7c7bb12abda8f11140d1b18c2b9b2e1e70", "domain": "thermal_transport", "formula_sha256": "2e54bbf0cc370a56c0ffd43f2ca8cdefa185eb02d8d2388124e6586960adac5c", "formula_latex": "{\\kappa}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "inputs": [ "HeatCapacity", "GroupVelocity", "MeanFreeDisplacement[bte_solver=direct_inverse]" ], "outputs": [ "ThermalConductivity[bte_solver=direct_inverse]" ], "binding_ids": [ "linearized-bte-conductivity" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "contract_kappa[bte_solver=rta]", "uid": "712b8145370a0072f225bd2f1d8c954b64311a3ca2f387320aa17e466926513c", "domain": "thermal_transport", "formula_sha256": "2e54bbf0cc370a56c0ffd43f2ca8cdefa185eb02d8d2388124e6586960adac5c", "formula_latex": "{\\kappa}_{\\alpha,\\beta} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {F}_{\\beta,\\mathbf{q},\\nu} {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "inputs": [ "HeatCapacity", "GroupVelocity", "MeanFreeDisplacement[bte_solver=rta]" ], "outputs": [ "ThermalConductivity[bte_solver=rta]" ], "binding_ids": [ "canonical-rta-thermal-conductivity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_kappa[transport_model=green_kubo]", "uid": "89d2d05c87fbf2327a442b230ebc1349fbcc1ec158fb59ea5ab22f7727d915f7", "domain": "thermal_transport", "formula_sha256": "e2bb1307e05a6ec0ade5c2361ced5a02baf17d92a07791dad85ed9215e77ea16", "formula_latex": "{\\kappa^{MD}}_{\\alpha,\\beta} = \\frac{V_{cell} \\int\\limits_{0}^{\\tau_{max}} {Jcorr}_{\\alpha,\\beta,\\tau}\\, d\\tau}{T^{2} k_{B}}", "inputs": [ "HeatCurrentACF", "Temperature" ], "outputs": [ "ThermalConductivity[transport_model=green_kubo]" ], "binding_ids": [ "md-correlation-estimators" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "contract_kappa[transport_model=hnemd]", "uid": "c3e43a0416eba1abf7d4cfa1467761af3b7aa7003df592db3a3599c635d58073", "domain": "thermal_transport", "formula_sha256": "0e78ac2ae21c0d6f4f24ed3ac8d60cf3870b5c0126e9ee7366e3e6371380d29b", "formula_latex": "{\\kappa^{MD}}_{\\alpha,\\beta} = \\frac{{J}_{\\alpha,t}}{T V_{cell} {F_{e}}_{\\beta}}", "inputs": [ "HeatCurrent", "Temperature" ], "outputs": [ "ThermalConductivity[transport_model=hnemd]" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "contract_kappa[transport_model=nemd]", "uid": "7b9e21caad74ed2be7cf49af55b1e745bb8a2606b035bee37c46c37a09cf58fc", "domain": "thermal_transport", "formula_sha256": "3b675fe9f2fa40bb78f74f19e7bb2e95561f224a18251a96172cd7e2a4267755", "formula_latex": "{\\kappa^{MD}}_{\\alpha,\\beta} = - \\frac{{J}_{\\alpha,t}}{{\\nabla T}_{\\beta}}", "inputs": [ "HeatCurrent", "Temperature" ], "outputs": [ "ThermalConductivity[transport_model=nemd]" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "contract_molar_entropy", "uid": "c9322877acc6349a7e18b29646674344b12d39565adf4ee3e23f2b82e1b53ab3", "domain": "thermal_transport", "formula_sha256": "dd26ff6b4ba598e3847d7cb5bea43db12244846b2adeb279f4a3a9d3d0df168f", "formula_latex": "S_{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {s}_{\\mathbf{q},\\nu}}{N_{q}}", "inputs": [ "Entropy" ], "outputs": [ "MolarEntropy" ], "binding_ids": [ "harmonic-joint-molar-ensemble", "harmonic-molar-entropy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_molar_free_energy", "uid": "9d4ded802829a8b2fe5544556eeb81534fe39e3ceae5efac0065e7ddcf3da152", "domain": "thermal_transport", "formula_sha256": "dbff2ebb2aece334d4cbf8a8ad1928d48e3660a69d64cb8b73746b54dedeb5b0", "formula_latex": "F_{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {f}_{\\mathbf{q},\\nu}}{N_{q}}", "inputs": [ "HelmholtzFreeEnergy" ], "outputs": [ "MolarHelmholtzFreeEnergy" ], "binding_ids": [ "harmonic-joint-molar-ensemble", "harmonic-molar-free-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_molar_heat_capacity", "uid": "00bbc32d579215c8ada01de36f6d5db89d0cd23ccd1e392bc4bc085e9c96fc8e", "domain": "thermal_transport", "formula_sha256": "d650d1a9d8bf0b675e7f416f15d6d29beda89179a7af59ab2afad6a1dd72e71c", "formula_latex": "C_V^{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {c}_{\\mathbf{q},\\nu}}{N_{q}}", "inputs": [ "HeatCapacity" ], "outputs": [ "MolarHeatCapacity" ], "binding_ids": [ "harmonic-joint-molar-ensemble", "harmonic-molar-heat-capacity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_molar_internal_energy", "uid": "77d1268cd072ac2acfbf95872ae6a8acc0a20ac9e196dfe7256048ad9f5f8f2d", "domain": "thermal_transport", "formula_sha256": "2df3e3d6b3c9796f181b1cd476cf85250ee59ca8d440c2a20770ef2bfe0d8065", "formula_latex": "E_{mol} = \\frac{N_{A} \\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {e}_{\\mathbf{q},\\nu}}{N_{q}}", "inputs": [ "InternalEnergy" ], "outputs": [ "MolarInternalEnergy" ], "binding_ids": [ "harmonic-joint-molar-ensemble", "harmonic-molar-internal-energy" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_molar_volume", "uid": "0438b30a3ed499df2f925e42d7b0ffc9bcff837ab8fdaa3f83985ce27c29421c", "domain": "thermodynamic_identities", "formula_sha256": "ebafbaa870d290d2cdac4a4f91bb5b8abb3501e83789d53a83dc8134a5d95638", "formula_latex": "V_{m} = N_{A} V_{cell}", "inputs": [], "outputs": [ "MolarVolume" ], "binding_ids": [], "status": "gap", "reason": "No reviewed Physlib model bridge is established here for the molar-volume producer; the Cp and thermal-Gruneisen identities and the thermoelectric mappings are separately classified." }, { "id": "contract_poisson_ratio", "uid": "1374919bed5b2ab25f700faf12ff8dc32b6fa3296fe92dd12092e3b9edd39e54", "domain": "mechanics", "formula_sha256": "b9c1551011306c38dd5511cafba2afc0051f45b276aa1f7b4e5d8d5a0228a49b", "formula_latex": "\\nu = \\frac{- 2 G + 3 K}{2 G + 6 K}", "inputs": [ "BulkModulus", "ShearModulus" ], "outputs": [ "PoissonRatio" ], "binding_ids": [ "linear-elasticity-voigt-moduli" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "contract_power_factor", "uid": "692d823ec06a56ba59453ff8650d602fc72d1aac991aae3e572b236e5a269306", "domain": "thermodynamic_identities", "formula_sha256": "0b12a94e400c0db0bfa5be59bc424b1869508be637badb889fb29d42fd6b39c7", "formula_latex": "PF = S^{2} \\sigma_{el}", "inputs": [ "ElectricalConductivity[carrier=electronic]", "SeebeckCoefficient" ], "outputs": [ "PowerFactor" ], "binding_ids": [ "thermoelectric-power-factor" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_pressure", "uid": "acbf580bd577b7499bc6806f2882511b90aaab32cabdb290a0de452d251e7514", "domain": "mechanics", "formula_sha256": "86f47f5fa90f900ae725b63de258ad5da3543e6ae3f07110e268241b7382edf3", "formula_latex": "P = \\frac{\\sum_{\\alpha=1}^{3} {\\sigma}_{\\alpha,\\alpha}}{3}", "inputs": [ "Stress" ], "outputs": [ "Pressure" ], "binding_ids": [ "continuum-stress-sign" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_shear_modulus", "uid": "3466399c40c467b80ca4b9b3a83c9c5b0d73df41ed0295e3a4d6685683972909", "domain": "mechanics", "formula_sha256": "a6d3c29f0a742deb9c3acf0be98694e08dd2ee4ce9e40e86b4bfb7d0d4931d69", "formula_latex": "G = - \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq \\beta \\leq 3}} {C}_{\\alpha,\\alpha,\\beta,\\beta}}{30} + \\frac{\\sum_{\\substack{1 \\leq \\alpha \\leq 3\\\\1 \\leq \\beta \\leq 3}} {C}_{\\alpha,\\beta,\\alpha,\\beta}}{10}", "inputs": [ "ElasticConstants" ], "outputs": [ "ShearModulus" ], "binding_ids": [ "linear-elasticity-voigt-moduli" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "contract_state_coefficient_of_variation", "uid": "28b7ede719aa19690afe0e07efd6fa73c49451f84edb7d126a8b78d07ce48894", "domain": "analog_device_metrology", "formula_sha256": "55d3fef6b0f4c399bb4e959c8f392959f84580e2182c4a50e9b4934a041267fb", "formula_latex": "c_{G} = \\frac{\\sigma_{G}}{G_{c}}", "inputs": [ "ConductanceState" ], "outputs": [ "StateCoefficientOfVariation" ], "binding_ids": [], "status": "gap", "reason": "A closed-form definition with nothing non-obvious to prove; no Lean theorem restates it, and it is not bound." }, { "id": "contract_thermal_gruneisen", "uid": "aa554ec7f8a638108f1039bffd5f1c727687f5fd0d018d61d2916e745a5f6f29", "domain": "quasiharmonic", "formula_sha256": "01d5dfc469084551b1508829d6a3c4d7f55bccf19207530724fb1f9a56137f26", "formula_latex": "\\gamma_{th} = \\gamma^{qha}{\\left(\\gamma_{G},\\omega \\right)}", "inputs": [ "Gruneisen", "Frequency" ], "outputs": [ "ThermalGruneisen" ], "binding_ids": [ "harmonic-qha-thermal-gruneisen-contraction" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_thermal_gruneisen_identity", "uid": "1348e12defce337587f9c2b152a3a1c88f9d27ec8cc48fc30167aebadd907043", "domain": "thermodynamic_identities", "formula_sha256": "d84872a220cba0af3fed426df7df8137e40c15bf547cdb3a4132d5e20af18f23", "formula_latex": "\\gamma_{th} = \\frac{K \\alpha_{V}}{C_V^{vol}}", "inputs": [ "ThermalExpansion", "BulkModulus", "VolumetricHeatCapacity" ], "outputs": [ "ThermalGruneisen" ], "binding_ids": [ "harmonic-static-thermal-gruneisen-identity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_volumetric_heat_capacity", "uid": "239c7dd3370e7b7e4456f1ade35c5d81edba69efda786d25e8e1e74f6fa297d9", "domain": "thermal_transport", "formula_sha256": "e89d3db8b665f634538055816c2e1336259f88bcd007badcf318c802d2b3ff92", "formula_latex": "C_V^{vol} = \\frac{\\sum_{\\substack{1 \\leq \\mathbf{q} \\leq N_{q}\\\\1 \\leq \\nu \\leq 3 N}} {c}_{\\mathbf{q},\\nu}}{N_{q} V_{cell}}", "inputs": [ "HeatCapacity" ], "outputs": [ "VolumetricHeatCapacity" ], "binding_ids": [ "harmonic-joint-volumetric-heat-capacity", "harmonic-volumetric-heat-capacity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "contract_youngs_modulus", "uid": "1b6ab3dbb0648151cd191a9a98d25d103e1a8461958a2f04c14dfa4627c248cf", "domain": "mechanics", "formula_sha256": "375c6fffee472f619a478562c1a12bc1fd48b7dc55a1a13abcffbaed8dfcc9e6", "formula_latex": "E_{Y} = \\frac{9 G K}{G + 3 K}", "inputs": [ "BulkModulus", "ShearModulus" ], "outputs": [ "YoungsModulus" ], "binding_ids": [ "linear-elasticity-voigt-moduli" ], "status": "local", "reason": "No current Physlib solid-elasticity/Voigt/EOS formula identification found. The bridge proves the signed inviscid stress/pressure contraction under its sign and unit conditions; identification with thermodynamic free-energy pressure remains unproved. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "contract_zt", "uid": "ff8a1d876f02c91d2c28a2ebecd11217c9496adc0b928b06ff13847bb6ac427b", "domain": "thermodynamic_identities", "formula_sha256": "261e8f1f0c68d1d4aca31e5984b597032ab4f2a3a3e0aa7b84beb577e095ae16", "formula_latex": "ZT = \\frac{PF T}{\\kappa_{tot}}", "inputs": [ "PowerFactor", "Temperature", "ThermalConductivity[contribution=total]" ], "outputs": [ "ZT" ], "binding_ids": [ "thermoelectric-figure-of-merit" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "depolarization_factors", "uid": "e8bd01e04d39ccad6f7046883bfa8e955e67c23fcc2f8868be045f327b18482f", "domain": "composites", "formula_sha256": "76cf8bc5f2cfa140cf9c69a817c7b4f0219d2b8336edf892f20184c6bfb304d4", "formula_latex": "L_{33} = \\begin{cases} \\frac{1}{3} & \\text{for}\\: p = 1 \\\\\\frac{- \\sqrt{1 - \\frac{1}{p^{2}}} + \\operatorname{atanh}{\\left(\\sqrt{1 - \\frac{1}{p^{2}}} \\right)}}{p^{2} \\left(1 - \\frac{1}{p^{2}}\\right)^{\\frac{3}{2}}} & \\text{for}\\: p > 1 \\\\\\frac{\\sqrt{-1 + \\frac{1}{p^{2}}} - \\operatorname{atan}{\\left(\\sqrt{-1 + \\frac{1}{p^{2}}} \\right)}}{p^{2} \\left(-1 + \\frac{1}{p^{2}}\\right)^{\\frac{3}{2}}} & \\text{otherwise} \\end{cases}", "inputs": [], "outputs": [ "DepolarizationFactor" ], "binding_ids": [ "composite-effective-medium" ], "status": "local", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "dibenedetto_glass_transition", "uid": "68e156cfe2da68f22c0e5c3c0bdc6426ce2fad2aa8a25714d90d79343d8d2ed4", "domain": "cure_kinetics", "formula_sha256": "6ec6864e62dbe6e996b088bcc17ed2c238b3839a5ca2ba8a30b6cdbd4f3ea03a", "formula_latex": "T_{g} = T_{g0} + \\frac{\\alpha_{cure} \\lambda_{DiB} \\left(- T_{g0} + T_{g\\infty}\\right)}{- \\alpha_{cure} \\left(1 - \\lambda_{DiB}\\right) + 1}", "inputs": [ "CureConversion" ], "outputs": [ "GlassTransitionTemperature" ], "binding_ids": [ "thermoset-cure-kinetics" ], "status": "local", "reason": "No matching current Physlib Kamal cure or DiBenedetto glass-transition model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "dissipate_pulse_energy", "uid": "3714b2588a2c9400c356080644dab2fcdf778f5224235d6389519e648f41ab8a", "domain": "analog_device_metrology", "formula_sha256": "e9880765ed8745f0415055ed80e1770d0fcf29a8935ec292448b5594058bb8e7", "formula_latex": "E_{pulse} = G_{c} V_{p}^{2} t_{p}", "inputs": [ "ConductanceState" ], "outputs": [ "ProgrammingPulseEnergy" ], "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "emit_thermionic_conductance", "uid": "58a959a412e7f92ec955cad34321812f5df453e9db8dcf4f8eb706f109bdf09a", "domain": "analog_device_metrology", "formula_sha256": "ae5ff88ffc8727046fd976c9097a5bb5745bb1908718b89aceb84914d38ed3da", "formula_latex": "G_{c} = \\frac{P_{th} T q_{e} e^{- \\frac{\\Phi_{Bn}}{T k_{B}}}}{k_{B}}", "inputs": [ "SchottkyBarrierHeight[band_carrier=electron]", "Temperature" ], "outputs": [ "ConductanceState" ], "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "evaluate_reservoir_classification", "uid": "557f968b85bdc45e519be10b1f5648d2d50005c4b18092f0435a0c74768a797a", "domain": "analog_device_metrology", "formula_sha256": "4be2b8ef7c0e9b2abcd4296f11fd776f7fe5b36a3fe42527e4ca6926a688bb1f", "formula_latex": "A_{cls} = F_{RC}{\\left(G_{c} \\right)}", "inputs": [ "ConductanceState" ], "outputs": [ "ClassificationAccuracy" ], "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "fit_arrhenius", "uid": "1038d2249c062baff931bbc06138496d90157bb4365334cae8c5092866e9b6a4", "domain": "materials", "formula_sha256": "30b75223fc1faba9136c4731d024ff4a258a6bb2ff8dcb7a170ffc25f9b633d5", "formula_latex": "D{\\left(T \\right)} = D_{0} e^{- \\frac{E_{a}}{T k_{B}}}", "inputs": [ "Diffusivity", "Temperature" ], "outputs": [ "ActivationEnergy" ], "binding_ids": [ "arrhenius-ionic-transport" ], "status": "local", "reason": "No matching current Physlib Einstein-MSD, Arrhenius, Nernst-Einstein, or cluster-expansion formula theorem identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "fourier_to_dos", "uid": "c142a31f3e5f89612d68319b4c2186372bf15592f88b09a1f2cc1356564428c0", "domain": "thermal_transport", "formula_sha256": "b4cdd53d454584e5202fb019a24ae3f4bc8ef51ba843fd0210c4e91456c43c13", "formula_latex": "{g}_{\\omega} = \\frac{6 \\int\\limits_{0}^{\\infty} \\cos{\\left(\\omega \\tau \\right)} {Cv}_{\\tau}\\, d\\tau}{\\pi {Cv}_{0}}", "inputs": [ "VelocityAutocorrelation" ], "outputs": [ "PhononDOS" ], "binding_ids": [ "vacf-spectrum-lorentzian-limit" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "hasselman_johnson", "uid": "33e0be2a3db8fa86dce200159c6c0a273161ee6d1a49c75a369b1621197e0a1a", "domain": "composites", "formula_sha256": "9d0e588f6e47a7bc1e8345b77f6593376a9a9d3a07f847fd9ab69b039db49d6b", "formula_latex": "\\kappa_{c} = \\frac{k_{m} \\left(2 f_{vol} \\left(- k_{m} + \\left(1 - \\frac{k_{m}}{G_{int} a_{rad}}\\right) {k_{f}}_{3,3}\\right) + 2 k_{m} + \\left(1 + \\frac{2 k_{m}}{G_{int} a_{rad}}\\right) {k_{f}}_{3,3}\\right)}{- f_{vol} \\left(- k_{m} + \\left(1 - \\frac{k_{m}}{G_{int} a_{rad}}\\right) {k_{f}}_{3,3}\\right) + 2 k_{m} + \\left(1 + \\frac{2 k_{m}}{G_{int} a_{rad}}\\right) {k_{f}}_{3,3}}", "inputs": [ "ThermalConductivity[role=matrix]", "ThermalConductivity[role=filler]", "InterfaceConductance", "FillerVolumeFraction" ], "outputs": [ "ThermalConductivity[effective_medium=nan,orientation=random]" ], "binding_ids": [ "composite-effective-medium" ], "status": "local", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "identity_dm", "uid": "e88e11c5770a9dc9c33df7139ddf40f072f3e415f20e15d860c3490104c3c34a", "domain": "thermal_transport", "formula_sha256": "2c152c985671629297ab2a041596f27c2b42870818f4eb36a96b7b25aa00387f", "formula_latex": "{D}_{i,j,\\mathbf{q}} = {D^{bare}}_{i,j,\\mathbf{q}}", "inputs": [ "BareDynamicalMatrix" ], "outputs": [ "DynamicalMatrix" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "kamal_cure_conversion", "uid": "0b2784cfd1c1758e66988322d83589700be6f7b1184c6f36a8d073eda43b57e1", "domain": "cure_kinetics", "formula_sha256": "f997bcdb5ce1ddd5b4061584d0e79463affd1ca1248131741721081bb20c7f00", "formula_latex": "t_{cure} = \\int\\limits_{\\alpha_{0,cure}}^{\\alpha_{cure}} \\frac{\\left(1 - \\alpha'\\right)^{- n_{cure}}}{A_{1,cure} e^{- \\frac{E_{1,cure}}{N_{A} T k_{B}}} + A_{2,cure} \\alpha'^{m_{cure}} e^{- \\frac{E_{2,cure}}{N_{A} T k_{B}}}}\\, d\\alpha'", "inputs": [ "Temperature" ], "outputs": [ "CureConversion" ], "binding_ids": [ "thermoset-cure-kinetics" ], "status": "local", "reason": "No matching current Physlib Kamal cure or DiBenedetto glass-transition model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "nan_effective_kappa", "uid": "04d65e9f7a2f612eb94d018ebde08e8a1b39c1c5754d03e785ad92991a909fda", "domain": "composites", "formula_sha256": "11db4a2afb71060dd77bbbb7da720b27898064b0068e327550d1435f1f85783a", "formula_latex": "\\kappa_{c} = \\frac{k_{m} \\left(f_{vol} \\left(\\frac{2 \\left(1 - L_{11}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right)}{L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right) + k_{m}} + \\frac{\\left(1 - L_{33}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}}\\right) + 3\\right)}{- f_{vol} \\left(\\frac{2 L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right)}{L_{11} \\left(- k_{m} + \\frac{{k_{f}}_{1,1}}{1 + \\frac{2 {k_{f}}_{1,1}}{G_{int} d_{1}}}\\right) + k_{m}} + \\frac{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}}\\right) + 3}", "inputs": [ "ThermalConductivity[role=matrix]", "ThermalConductivity[role=filler]", "InterfaceConductance", "FillerVolumeFraction", "DepolarizationFactor" ], "outputs": [ "ThermalConductivity[effective_medium=nan,orientation=random]" ], "binding_ids": [ "composite-effective-medium" ], "status": "local", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "nan_effective_kappa_aligned", "uid": "e86ac8a4b69dbb670dbe3f8ec7fda7ff0309670bac4713851ced64a061cd7962", "domain": "composites", "formula_sha256": "78db51dc39f38b3dff3f71c49db5450b2c9e89907cbbd61ba5b01852433353fb", "formula_latex": "\\kappa_{c} = \\frac{k_{m} \\left(\\frac{f_{vol} \\left(1 - L_{33}\\right) \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}} + 1\\right)}{- \\frac{L_{33} f_{vol} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right)}{L_{33} \\left(- k_{m} + \\frac{{k_{f}}_{3,3}}{1 + \\frac{2 {k_{f}}_{3,3}}{G_{int} d_{3}}}\\right) + k_{m}} + 1}", "inputs": [ "ThermalConductivity[role=matrix]", "ThermalConductivity[role=filler]", "InterfaceConductance", "FillerVolumeFraction", "DepolarizationFactor" ], "outputs": [ "ThermalConductivity[effective_medium=nan,orientation=aligned]" ], "binding_ids": [ "composite-effective-medium" ], "status": "local", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "propagate_programming_error", "uid": "4517674f602a29f64ecc7140384fc486ee295233ce03ca7dc6bbec7eb00a91c2", "domain": "analog_device_metrology", "formula_sha256": "4977f8d380bdaa87a5000420f6c515cae3a2203ab4269d84301a6252b65c8917", "formula_latex": "\\epsilon_{dot} = \\frac{c_{G} \\sqrt{\\frac{S_{1} w_{max}}{W_{G} - 1} + \\frac{S_{2}}{2} + \\frac{n_{w} w_{max}^{2}}{2 \\left(W_{G} - 1\\right)^{2}}}}{S_{1}}", "inputs": [ "StateCoefficientOfVariation", "ConductanceWindow" ], "outputs": [ "DotProductError" ], "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "provide_born_charges", "uid": "8e585d905e8a35061ccdc51d553ffce2940699c2159a975c75aeff9abba876f4", "domain": "thermal_transport", "formula_sha256": "608271f922f98a94ecf89b9e1b91ea874697ec23fdfa6a096f2b66aaf2ff8afa", "formula_latex": "{Z^{*}}_{i,\\gamma,\\alpha} = Z^*_{provided}", "inputs": [], "outputs": [ "BornCharges" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "provide_dielectric_tensor", "uid": "64badcbbde76ff0975a351c9de13aa12ca735ce7d3e986723d5cf19b7f1125a1", "domain": "thermal_transport", "formula_sha256": "639a555002cc7aa115d3bba042d932a03a640372f1dfa7f60621e3d30d5de95a", "formula_latex": "{\\varepsilon_{\\infty}}_{\\gamma,\\delta} = \\varepsilon_{\\infty,provided}", "inputs": [], "outputs": [ "DielectricTensor" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "provide_isotope_abundances", "uid": "c4f3608ad7e68c033d01ba334a756592961af7f922d496a4cf13c809677ccaf4", "domain": "thermal_transport", "formula_sha256": "2098e2119d74db36c5d4c49819c4283ac288df63892197e7310e3a8b86eff73c", "formula_latex": "{g}_{i} = g_{provided}", "inputs": [], "outputs": [ "IsotopeAbundances" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "provide_potential", "uid": "c6655ff437e25290b3f7a8181260299ecd1839430b5156a78b24d829db82e45d", "domain": "thermal_transport", "formula_sha256": "024817078a6f5f0055a969533961879bba04da83688e6ffa6626cdb582eccab2", "formula_latex": "V{\\left(\\{u\\} \\right)} = V_{\\mathrm{provided}}", "inputs": [], "outputs": [ "Potential" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "provide_temperature", "uid": "7e819af4e68ff4465eb8461e12d5d81bd6dbcb395280896a38e5cf4244642737", "domain": "thermal_transport", "formula_sha256": "cc12c4f5373c62f73b7a0d5fbd0c30f5c6c6191bedf96d58f395c3508a3e798f", "formula_latex": "T = T_{\\mathrm{provided}}", "inputs": [], "outputs": [ "Temperature" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "provide_transmission", "uid": "f4a8562c1bf037775616e93ace8d78be9cda7338810339c51a41a2662c756079", "domain": "thermal_transport", "formula_sha256": "ce516ddd46f59f03a7fe8f547f252995b89d489522ad4726e44c026ac3de1910", "formula_latex": "{\\mathcal{T}}_{\\omega} = \\mathcal{T}_{provided}", "inputs": [], "outputs": [ "PhononTransmission" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "ramp_set_voltage", "uid": "9ab6278d7c06fcac5a8f9bcea7c65eb193c793c435822f24ca6762da34880dea", "domain": "analog_device_metrology", "formula_sha256": "dd1bf4e3e7338f03029e6d92efb5631aed804fe383d0cf857b8fb26703ce77a9", "formula_latex": "V_{set} = - \\frac{\\gamma T k_{B}}{\\alpha_{F}} + \\frac{E_{m} - T k_{B} \\log{\\left(\\frac{T g_{r} k_{B} \\nu_{0}}{\\alpha_{F}} \\right)}}{\\alpha_{F}}", "inputs": [ "MigrationBarrier", "Temperature" ], "outputs": [ "SetVoltage" ], "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "resolve_effective_conductivity", "uid": "944a08d41ead4ae8f8b899345c3c046d36fb42fe57b5a62bd17a117de5388738", "domain": "composites", "formula_sha256": "b5bea41b6c623f7c09f1bf24dcae58ebab3c0cdd90ad966bc43a45b44867e12b", "formula_latex": "\\text{True}", "inputs": [ "ThermalConductivity[effective_medium=nan,orientation=random]" ], "outputs": [ "ThermalConductivity" ], "binding_ids": [], "status": "gap", "reason": "No matching current Physlib Nan/Hasselman-Johnson/depolarization/interfacial NEMD formula model identified." }, { "id": "resolve_thermal_conductivity", "uid": "516d748ca6d647abd4346336058ba9c42970078bee2991aa45cf8d4cfb8ff49d", "domain": "thermal_transport", "formula_sha256": "b5bea41b6c623f7c09f1bf24dcae58ebab3c0cdd90ad966bc43a45b44867e12b", "formula_latex": "\\text{True}", "inputs": [ "ThermalConductivity[bte_solver=direct_inverse]" ], "outputs": [ "ThermalConductivity" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "run_md", "uid": "2444a4dc420004700fd7fe2e6b0f76ad69ac8799c7d8eb3ce2186e6ea138045b", "domain": "thermal_transport", "formula_sha256": "c10514538935e1d22aeaa65363aa96e2585b5354e31283ae27458b38a868cda4", "formula_latex": "{r}_{i,\\alpha,t + 1} = \\frac{\\Delta t^{2} {F^{md}}_{i,\\alpha,t}}{2} + \\Delta t {v}_{i,\\alpha,t} + {r}_{i,\\alpha,t}", "inputs": [ "Potential", "Temperature" ], "outputs": [ "Trajectory" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "sample_pmf", "uid": "40c9c4bdbf9b62a690eceb9042667a3ebc6c73a53ac141b538c0d2e44037d2a1", "domain": "thermal_transport", "formula_sha256": "96d7b1f49c9358f82e36d94849e0e68e46ccccf786493c282630d47a80d0d19e", "formula_latex": "F = F_{pmf}{\\left(r,T \\right)}", "inputs": [ "Trajectory", "Temperature" ], "outputs": [ "PotentialOfMeanForce" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "sample_quantum_heat_capacity", "uid": "8b0442bc348b8bb2e337fa19c06e6ab2e1c3568515b4322c834cbf197207469c", "domain": "thermal_transport", "formula_sha256": "e6fc8fccecebc0ecc5216a20e9c2d8f4bd0d35c2970b024ecddb2c2ae0009dfb", "formula_latex": "C_{V} = C_V^{pimd}{\\left(r,T \\right)}", "inputs": [ "Trajectory", "Temperature" ], "outputs": [ "HeatCapacity[method=pimd]" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "sample_quantum_kinetic_energy", "uid": "9fc76c80bba656236591522ff1955da5d9c313be5d0bc5beca5db579022d2ccd", "domain": "thermal_transport", "formula_sha256": "055dc55bf27dc0811d5af54e2159d0f9857022c78b30fdc37060b4ed5fc6a991", "formula_latex": "E_{K} = E_K^{cv}{\\left(r,T \\right)}", "inputs": [ "Trajectory", "Temperature" ], "outputs": [ "QuantumKineticEnergy" ], "binding_ids": [], "status": "gap", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas." }, { "id": "schottky_mott_barrier", "uid": "5495d20eea478bf7e68e39211a5881ed9e1ba5bb0d8bda8d0b943c03e69a9a6c", "domain": "analog_device_metrology", "formula_sha256": "0e465ee0e2a48215b11ac34d2378b111efd8d2a07962bf3fc3be0ee58b49b7b0", "formula_latex": "\\Phi_{Bn} = \\Phi_{W} - \\chi_{s}", "inputs": [ "WorkFunction", "ElectronAffinity" ], "outputs": [ "SchottkyBarrierHeight[band_carrier=electron]" ], "binding_ids": [], "status": "gap", "reason": "Phi_Bn = Phi_W - chi is a difference with nothing to prove algebraically; its validity (no Fermi-level pinning and no interface dipole, while measured pinning factors are about 0.1 for Si, GaAs and MoS2) cannot be established algebraically. Catalog defect: WorkFunction does not distinguish a metal from a semiconductor, so a semiconductor's own work function can compose into a SchottkyBarrierHeight." }, { "id": "solve_bte[bte_solver=direct_inverse]", "uid": "3e5e0cdbda5333384f4df3d6b4e517854038fea4377f64e7fed212b9d995d2ea", "domain": "thermal_transport", "formula_sha256": "8ee139deb4c747f5b98fc57ede932ac9471296bb7fce4f9d04cd4d6dc0feafcc", "formula_latex": "\\sum_{\\substack{1 \\leq \\mathbf{q'} \\leq N_{q}\\\\1 \\leq \\nu' \\leq 3 N}} {F}_{\\alpha,\\mathbf{q'},\\nu'} {\\mathcal{M}}_{\\mathbf{q},\\nu,\\mathbf{q'},\\nu'} = {c}_{\\mathbf{q},\\nu} {v}_{\\alpha,\\mathbf{q},\\nu}", "inputs": [ "Frequency", "Eigenvectors", "ForceConstants[order=3]", "GroupVelocity", "HeatCapacity", "Linewidth[channel=total]", "Temperature" ], "outputs": [ "MeanFreeDisplacement[bte_solver=direct_inverse]" ], "binding_ids": [ "linearized-bte-conductivity" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "solve_bte[bte_solver=rta]", "uid": "d1c7da072f801890c5b4d17bff8f196b38823ae46102eb308c2c50438edf8b19", "domain": "thermal_transport", "formula_sha256": "0b94f07d722c2b4c036e62b1790aa56e7427ffdd433bb5458e23219e6d9b923c", "formula_latex": "{F}_{\\alpha,\\mathbf{q},\\nu} = \\frac{{v}_{\\alpha,\\mathbf{q},\\nu}}{2 {\\Gamma}_{\\mathbf{q},\\nu}}", "inputs": [ "Frequency", "GroupVelocity", "Linewidth[channel=total]", "Temperature" ], "outputs": [ "MeanFreeDisplacement[bte_solver=rta]" ], "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "solve_equilibrium", "uid": "8e00f3c45c66888481ee6a2004e915534c06dfbc94ef59c7c761b85a1a7365a8", "domain": "thermochemistry", "formula_sha256": "e462f6823acba4718605093dd6a9c2f5b65d76e52a2f0f936f34b1d30f06d951", "formula_latex": "G_{m} = G^{\\min}{\\left(\\mathcal{D},T \\right)}", "inputs": [ "AssessedDatabase", "Temperature" ], "outputs": [ "MolarGibbsEnergy" ], "binding_ids": [ "calphad-binary-equilibrium" ], "status": "local", "reason": "No current Physlib CALPHAD/Gibbs-assessed database or per-mole-atoms formula correspondence identified; Helmholtz is not Gibbs. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "solve_ground_state", "uid": "f2e7f20060fc7a467fa60d0dcede02095054636adf260b0f78f1206350524738", "domain": "dft_ground_state", "formula_sha256": "2016eb01f5d527d8d736111431098456caa9df8ba8fe52134e480e12abd5de5d", "formula_latex": "E_{tot} = E_{KS}{\\left(\\mathcal{S},V \\right)}", "inputs": [ "Structure", "Potential" ], "outputs": [ "TotalEnergy" ], "binding_ids": [], "status": "gap", "reason": "No current Physlib Kohn-Sham, DFT, atomistic derivative, or ground-state producer formula identification found in the audited source." }, { "id": "subthreshold_conductance_tangent_time", "uid": "f98ab6a713f029111d0c63c1017b87934ee2d50296f1e0a49289217cb20495eb", "domain": "analog_device_metrology", "formula_sha256": "461ebabd3da525239d99ef87d26a6dbe8324ba981c72301c83b44e85a0887b3b", "formula_latex": "\\tau_{G} = \\frac{\\tau_{it}}{\\log{\\left(\\frac{G_{c}}{G_{eq}} \\right)}}", "inputs": [ "InterfaceTrapRelaxationTime", "ConductanceState" ], "outputs": [ "ConductanceRelaxationTime" ], "binding_ids": [], "status": "gap", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention." }, { "id": "sum_linewidths", "uid": "895d9e39c50870b6929c98b38c37b813f0688f3ade6b09e72c1bf1fad683ab7c", "domain": "thermal_transport", "formula_sha256": "be92953f056d5d9606396776a4b777bcf1806a44aea1dcf0e82ea7527dcbf85b", "formula_latex": "{\\Gamma^{tot}}_{\\mathbf{q},\\nu} = {\\Gamma^{anh}}_{\\mathbf{q},\\nu} + {\\Gamma^{bnd}}_{\\mathbf{q},\\nu} + {\\Gamma^{iso}}_{\\mathbf{q},\\nu}", "inputs": [ "Linewidth[channel=anharmonic_3ph]", "Linewidth[channel=isotope]", "Linewidth[channel=boundary]" ], "outputs": [ "Linewidth[channel=total]" ], "binding_ids": [ "phonon-scattering-channels" ], "status": "local", "reason": "The compiled lattice-dynamics bridges cover the crystal dynamical matrix, the dispersion relation at commensurate wavevectors, the group velocity on differentiable eigenvector paths and the relaxation-time contraction under explicit model conditions. No current Physlib phonon scattering, anharmonic, numerical estimator, molecular-dynamics or Green-Kubo theorem was identified for the remaining formulas. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." }, { "id": "sum_thermal_conductivity", "uid": "e5d0388c6c18b3fb6cb08647ddaf431bfb39e60f33e0abd1ecdd2cb0c05575ee", "domain": "thermodynamic_identities", "formula_sha256": "05612b74df513ca263a0cbb8a80f9c5f703dceced4703b2bb1f8adf76ff8f28e", "formula_latex": "\\kappa_{tot} = \\kappa + \\kappa_{e}", "inputs": [ "ThermalConductivity[bte_solver=direct_inverse]", "ElectronicThermalConductivity" ], "outputs": [ "ThermalConductivity[contribution=total]" ], "binding_ids": [ "thermoelectric-total-conductivity" ], "status": "mapped", "reason": "See the conditions and evidence for each mapping." }, { "id": "zero_bias_retention", "uid": "10e6c7d5ab4087efdea29cf2219771cd55f19a87c50c19ca46756620116344e8", "domain": "analog_device_metrology", "formula_sha256": "0d49b118dcf2f3478766bb23218c8161e098df186770bcebe8779d0aa58aaf61", "formula_latex": "\\tau_{ret} = \\frac{e^{\\frac{E_{m}}{T k_{B}}}}{\\nu_{0}}", "inputs": [ "MigrationBarrier", "Temperature" ], "outputs": [ "RetentionTime" ], "binding_ids": [ "analog-device-metrology" ], "status": "local", "reason": "No matching current Physlib model was identified for the reviewed device-metrology formulas: conductance, drift, programming error, thermionic and Schottky emission, ramp, and retention. A local Lean proof over Mathlib covers it under that binding's conditions; see its conditions and evidence." } ] }, "coverage": { "nodes_total": 136, "nodes_mapped": 35, "operators_total": 133, "operators_mapped": 29, "nodes_local": 44, "operators_local": 56, "bindings_total": 54, "source_correspondences": 5, "compiled_bridges": 49, "nodes_compiled": 26, "nodes_compiled_outputs": 25, "nodes_compiled_interpretations": 1, "operators_compiled": 29, "operators_compiled_local": 56, "nodes_compiled_local": 49, "operators_proved": 85, "nodes_proved": 75, "upstream_declarations_referenced": 36, "upstream_modules_referenced": 10, "relations": { "exact_definition": 1, "conditional": 29, "conceptual": 4, "local": 20 } } }