ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Different Fourier coefficients give different model fluxes

Physical scope

Comparison of two stipulated isotropic Fourier models on the same temperature profile; no material measurement or universal constitutive law is asserted.

Example

On $O = (0, L)^{3}$ in metre coordinates and $I = (0, τ)$ in second coordinates, with $L > 0 m$ and $τ > 0 s$ , prescribe $T (x, t) = T_{0} + Δ T x_{1} / L$ , where $T_{0} > 0 K$ and $Δ T > 0 K$ . In the chosen Cartesian coordinates, define $e_{1} : = (1, 0, 0) \in R^{3}$ . Compare two isotropic model choices $κ_{1}, κ_{2} > 0$ , $κ_{1} \neq κ_{2}$ , each using $j_{Q} = - κ_{i} \nabla T$ . The coefficients have units $W m^{- 1} K^{- 1}$ .

Facts & Assumptions

Given: The Fourier-model definition and the stipulated fields and coefficients.

[F1]

Scope of transport constitutive laws: for a temperature field with continuous first spatial partial derivatives, $\nabla T = (\partial_{x_{1}} T, \partial_{x_{2}} T, \partial_{x_{3}} T)$ in the physical one-based coordinate display (function indices $i - 1$ ), and a Fourier model may postulate $j_{Q} = - K \nabla T$ , where $K$ has entries in $W m^{- 1} K^{- 1}$ ; its isotropic form is $K = κ I_{3}$ with $κ \geq 0$ .

[A1]

$T_{0} > 0 K$ , $Δ T > 0 K$ , $L > 0 m$ ; $e_{1} = (1, 0, 0)$ in the chosen Cartesian coordinates; $κ_{1}, κ_{2} > 0$ in $W m^{- 1} K^{- 1}$ with $κ_{1} \neq κ_{2}$ ; both models use the same profile $T = T_{0} + Δ T x_{1} / L$ ; and model $i$ adopts $j_{Q, i} = - κ_{i} \nabla T$ .

Verification

Given: [F1] and [A1].

1.1A1constructalgebra

The common temperature gradient is $\nabla T = (Δ T / L) e_{1}$ . The two assigned model fluxes are therefore $j_{Q, 1} = - κ_{1} (Δ T / L) e_{1}$ and $j_{Q, 2} = - κ_{2} (Δ T / L) e_{1}$ .

2.1A1step 1.1algebra

Their difference is $j_{Q, 1} - j_{Q, 2} = - (κ_{1} - κ_{2}) (Δ T / L) e_{1} \neq 0$ . The same field profile therefore has different predicted fluxes under these two constitutive choices.

3.1F1A1step 2.1∎

The calculation uses the Fourier relation as a stated model input and does not determine either $κ_{i}$ from equilibrium entropy data. It is a model comparison, not a measured result.

Depends on

Scope of transport constitutive laws

Different Fourier coefficients give different model fluxes

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels