CounterexampleConstruction: 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)

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Equal branch volumes do not give a Clapeyron graph

Physical scope

A constructed pair of smooth one-component Gibbs charts with zero molar-volume jump; it shows why the local nonzero-jump hypothesis cannot be omitted and makes no material claim.

Statement refuted

Equality of two smooth one-component molar Gibbs branches near a coexistence point always defines a unique differentiable pressure graph $p = p (T)$ on an open temperature interval, with slope $Δ s / Δ v$ , even when the molar-volume jump $Δ v$ vanishes.

Example

Fix $T_{0} > 0$ , $p_{0} > 0$ , $g_{0} \in R$ , $v_{0}, c, κ > 0$ , and distinct positive molar entropies $s_{α}, s_{β}$ . Use units $g_{0} \in J m o l^{- 1}$ , $v_{0} \in m^{3} m o l^{- 1}$ , $c \in J m o l^{- 1} K^{- 1}$ , and $κ \in m^{3} m o l^{- 1} P a^{- 1}$ . On a common open chart define

g_{i} (T, p) = g_{0} - s_{i} (T - T_{0}) - \frac{c}{2 T_{0}} (T - T_{0})^{2} + v_{0} (p - p_{0}) - \frac{κ}{2} (p - p_{0})^{2},

with $i \in {α, β}$ . Let $G_{i} = n_{i} g_{i}$ and adopt $S_{i} = - \partial_{T} G_{i}$ , $V_{i} = \partial_{p} G_{i}$ , $μ_{i} = \partial_{n_{i}} G_{i}$ , and $U_{i} = G_{i} + T S_{i} - p V_{i}$ . This gives same-tag scaling and smooth local Gibbs charts. At $(T_{0}, p_{0})$ the molar Gibbs values agree and the molar-volume jump is zero; the entropies differ.

Facts & Assumptions

Given: The two smooth branch Gibbs functions and conjugate assignments above, with $Δ s = s_{β} - s_{α} \neq 0$ .

[A1]

Constructed local Gibbs model: The branch functions and natural conjugates are exactly those declared in the Example; no stable-endpoint or material-preparation claim is made.

[F1]

Local Clapeyron slope on a regular pure-phase coexistence branch: The local coexistence graph and slope conclusions require $Δ v \neq 0$ .

[L1]

The Euclidean inverse function theorem: A C1 map between open Euclidean neighborhoods with invertible derivative at a point has a local C1 inverse.

Counterexample

1.1givenA1L1algebra

At $(T_{0}, p_{0})$ , differentiation gives $S_{i} / n_{i} = s_{i}$ , $V_{i} / n_{i} = v_{0}$ , and $μ_{i} = g_{i} = g_{0}$ . Also $U_{i, T} = n_{i} c$ , $V_{i, p} = - n_{i} κ$ , and $V_{i, T} = 0$ , so the derivative determinant for $(T, p) \mapsto (U_{i}, V_{i})$ at fixed positive amount is $- n_{i}^{2} c κ \neq 0$ . Thus [L1] supplies valid local state charts. The branch equality function is

F (T, p) : = g_{β} - g_{α} = - (s_{β} - s_{α}) (T - T_{0}),

whose zero set is $T = T_{0}$ with every nearby chart pressure permitted.

2.1givenF1step 1.1algebra∎

Here $Δ v = v_{β} - v_{α} = 0$ , so the zero set is not the graph of a unique pressure as a function of temperature on an open interval about $T_{0}$ , and the quotient $Δ s / Δ v$ is undefined. The nonzero-jump hypothesis of [F1] fails, so this model does not contradict that theorem.

Equal branch volumes do not give a Clapeyron graph

Statement refuted

Example

Facts & Assumptions

Counterexample

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