Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-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.

Differentials of the thermodynamic potentials

Physical scope

Exact local differential identities on positive-temperature, C2, single-phase energy-representation charts with the stated Legendre coordinate maps invertible; no identity is asserted at singular response charts or phase boundaries.

Statement

On a valid local energy-representation chart from Local energy representation and thermodynamic conjugates, let $s, V, n_{j}, U, T, p, μ_{j}$ and the potentials $H, F, G, Ω$ be as defined in Local thermodynamic Legendre potentials. On every open neighborhood where the corresponding natural-coordinate maps have their stated $C^{1}$ inverses, the potentials are C2 and their differentials are

$d U = T d s - p d V + \sum_{j = 1}^{r} μ_{j} d n_{j}, d H = T d s + V d p + \sum_{j = 1}^{r} μ_{j} d n_{j},$

$d F = - s d T - p d V + \sum_{j = 1}^{r} μ_{j} d n_{j}, d G = - s d T + V d p + \sum_{j = 1}^{r} μ_{j} d n_{j},$

$d Ω = - s d T - p d V - \sum_{j = 1}^{r} n_{j} d μ_{j} .$

All differentials are local state-function differentials. They do not assert that heat or work is an exact differential on process space.

Facts & Assumptions

Given: A fixed constitution $c$ , an open C2 single-phase chart in coordinates $(s, V, n_{1}, \dots, n_{r})$ , the local energy representation, and the valid natural-coordinate inverse charts named in the Statement. Use SI units: $U, H, F, G, Ω$ in joules; $s$ in $J K^{- 1}$ ; $T$ in kelvins; $V$ in $m^{3}$ ; $p$ in pascals; $n_{j}$ in moles; and $μ_{j}$ in $J m o l^{- 1}$ .

[F1]

Local energy representation and thermodynamic conjugates: Its Statement gives $U_{c, s} = T_{c}$ , $U_{c, V} = - p_{c}$ , and $U_{c, n_{j}} = μ_{c, j}$ on a local C2 chart. The carrier is registered in batch 8; its current proof uses the C1 graph interface of the batch-4 draft thm-tdmath-local-implicit-rank. That supplier's Step 1 readiness record is authoring status, not independent proof acceptance, so this use remains conditional on its mathematical review.

[F2]

Local thermodynamic Legendre potentials: $H = U + p V$ , $F = U - T s$ , $G = F + p V$ , and $Ω = F - \sum_{j} μ_{j} n_{j}$ on the stated inverse charts.

[F3]

Differential of a C2 partial Legendre transform on a local inverse chart: For a C2 scalar function and an assumed C1 local inverse of its conjugate-coordinate map, the partial Legendre transform has differential $- u \cdot d x + D_{z} f \cdot d z$ .

Proof

1.1givenF1algebra

By [F1], the coordinate derivative of $U$ is $d U = U_{s} d s + U_{V} d V + \sum_{j} U_{n_{j}} d n_{j} = T d s - p d V + \sum_{j} μ_{j} d n_{j}$ . This step is conditional on the actual supplier interface and proof status recorded in [F1].

2.1F2F3step 1.1

Apply [F3] to the $V$ transform with conjugate coordinate $- p = U_{V}$ and to the $s$ transform with conjugate coordinate $T = U_{s}$ . With the signs fixed in [F2], this gives $d H = T d s + V d p + \sum_{j} μ_{j} d n_{j}$ and $d F = - s d T - p d V + \sum_{j} μ_{j} d n_{j}$ .

2.2F2F3step 1.1

Apply [F3] to the joint $(s, V)$ transform with conjugates $(T, - p)$ . The definition [F2] has $G = U - T s + p V$ , so the transform differential is $d G = - s d T + V d p + \sum_{j} μ_{j} d n_{j}$ .

3.1F2F3step 1.1∎

Apply [F3] to the joint $(s, n)$ transform with conjugates $(T, μ)$ . The definition [F2] has $Ω = U - T s - \sum_{j} μ_{j} n_{j}$ , so the transform differential is $d Ω = - s d T - p d V - \sum_{j} n_{j} d μ_{j}$ .

Remarks

The argument is a complete conditional deduction from the current local-energy statement. Its chart and derivative interface match the use. The supplier's proof remains a draft conditional on the registered batch-4 implicit-rank theorem; neither supplier readiness record is independent proof acceptance. This proof-status gap is recorded in batch 9 readiness evidence.

Differentials of the thermodynamic potentials

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources