ExampleConstruction: AI-adaptedVerification: 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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One component in two phases has one regular equation-set degree of freedom

Physical scope

A constructed pair of smooth one-component Gibbs charts; the example verifies the regular equation-set rank and dimension only, and does not assert a stable or realized coexistence endpoint.

Example

Let $C = 1$ , $P = 2$ , and $R = 0$ . Fix $T_{0} = 300 K$ and $p_{0} = 10^{5} P a$ , and take a common open chart around $(T_{0}, p_{0})$ . On it define the branch Gibbs potentials per mole

g_{1} (T, p) = g_{0} - s_{0} (T - T_{0}) - \frac{c}{2 T_{0}} (T - T_{0})^{2} + v_{1} (p - p_{0}) - \frac{κ}{2} (p - p_{0})^{2}, g_{2} (T, p) = g_{0} - s_{0} (T - T_{0}) - \frac{c}{2 T_{0}} (T - T_{0})^{2} + v_{2} (p - p_{0}) - \frac{κ}{2} (p - p_{0})^{2},

where $g_{0}$ is in $J m o l^{- 1}$ , $s_{0}, c > 0$ are in $J m o l^{- 1} K^{- 1}$ , $v_{1}, v_{2} > 0$ are distinct molar volumes in $m^{3} m o l^{- 1}$ , and $κ > 0$ is in $m^{3} m o l^{- 1} P a^{- 1}$ . Set $G_{i} = n_{i} g_{i}$ with natural Gibbs assignments $S_{i} = - \partial_{T} G_{i}$ , $V_{i} = \partial_{p} G_{i}$ , and $μ_{i} = \partial_{n_{i}} G_{i}$ ; define $U_{i} = G_{i} + T S_{i} - p V_{i}$ . The derivative-defined chemical potentials are $μ_{i} = g_{i}$ , and they agree at $(T_{0}, p_{0})$ . This is a candidate equation-map solution, not a claim of entropy-maximizing coexistence.

For each branch, fix a reference amount $n_{*} > 0$ in moles and an open interval $J_{n} \subset (0, \infty)$ around it. Initially take $O = O_{T p} \times J_{n}$ , where $O_{T p}$ is the displayed open $(T, p)$ chart about $(T_{0}, p_{0})$ , and shrink it about $(T_{0}, p_{0}, n_{*})$ to the inverse-function neighborhood verified below, retaining $T, p, n > 0$ . The full derivative of $(T, p, n) \mapsto (U, V, n)$ is block triangular with the last diagonal entry $1$ , so the displayed fixed-amount determinant is also its full determinant. The constructed branch's local physical state map is $χ : (T, p, n) \mapsto (U, V, n)$ (using the positive amount and interior composition coordinates displayed here). On the inverse patch verified below, define its numerical domain $O_{c} = Φ_{c}^{- 1} (χ (O))$ and declare this patch to be the local model's $D_{c}$ patch, where $Φ_{c} (\hat{U}, \hat{V}, \hat{n}) = (\hat{U} J, \hat{V} m^{3}, \hat{n} m o l)$ is the unit map of Entropic temperature, pressure, and molar chemical potentials. Define the numerical entropy by $S_{c} = S^{S I} \circ Φ_{c}$ , with $S^{S I}$ supplied by the Gibbs assignments through this inverse. Thus the displayed derivatives are derivatives in physical SI coordinates on $Φ_{c} (O_{c})$ ; the model patch is not an empirical claim about an independently specified material domain.

Facts & Assumptions

Given: The two Gibbs branches and the open $(T, p)$ chart described above, with $v_{1} \neq v_{2}$ . The example is a conditional equation-map model, not an experimental claim.

[A1]

The common positive-temperature, positive-pressure-or-real-pressure chart has no independent composition coordinates because $C = 1$ .

[A2]

Adopted local Gibbs model: $G_{i} = n_{i} g_{i}$ , $S_{i} = - \partial_{T} G_{i}$ , $V_{i} = \partial_{p} G_{i}$ , $μ_{i} = \partial_{n_{i}} G_{i}$ , and $U_{i} = G_{i} + T S_{i} - p V_{i}$ .

[F1]

Regular rank form of the finite-component phase dimension count: For $C = 1$ , $P = 2$ , $R = 0$ , a coexistence-equation zero with a nonsingular derivative minor has local equation-set dimension $C - P + 2 = 1$ .

[F2]

The model and calculation are only for a candidate equation-map solution; no stable or realized coexistence endpoint is assumed.

[L1]

The Euclidean inverse function theorem: A $C^{1}$ map between open Euclidean sets with invertible derivative at a point has a local $C^{1}$ inverse there.

[L2]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ supplies the differential chain rule; $C^{k}$ Euclidean maps are closed under componentwise algebra and composition preserves $C^{1}$ under finite algebra and composition. Continuity and derivatives of positive-base real powers gives $(t^{- 1})^{'} = - t^{- 2}$ on $t > 0$ , whose continuous derivative makes this reciprocal $C^{1}$ .

Verification

1.1givenA2L1algebra

From [A2], at $(T_{0}, p_{0})$ , $S_{i} = n_{i} s_{0}$ , $V_{i} = n_{i} v_{i}$ , $U_{i, T} = n_{i} c$ , $V_{i, p} = - n_{i} κ$ , and $V_{i, T} = 0$ . Thus the $(T, p)$ derivative matrix of $(U_{i}, V_{i})$ at fixed amount has determinant $- n_{i}^{2} c κ \neq 0$ . The full $(T, p, n) \mapsto (U_{i}, V_{i}, n)$ Jacobian has this same determinant, since its last row is $(0, 0, 1)$ . Thus [L1] supplies the claimed open full-state inverse patch about $(T_{0}, p_{0}, n_{*})$ .

2.1givenA2L2step 1.1algebra

On that patch the Gibbs assignments and the chain rule give $d G_{i} = - S_{i} d T + V_{i} d p + g_{i} d n$ and hence $d U_{i} = T d S_{i} - p d V_{i} + g_{i} d n$ . Pulling these identities through the $C^{1}$ inverse, $d S_{i} = T^{- 1} d U_{i} + (p / T) d V_{i} - (g_{i} / T) d n$ . Each coefficient is $C^{1}$ by [L2], since the inverse coordinates and $g_{i}$ are $C^{1}$ and $T > 0$ . Thus the constructed entropy is $C^{2}$ , with $S_{i, U} = 1 / T > 0$ , $S_{i, V} = p / T$ , and $S_{i, n} = - g_{i} / T$ , so its derivative-defined intensive quantities reproduce the Gibbs assignments.

3.1givenA2step 1.1step 2.1algebra

The only coexistence equation is $F (T, p) : = g_{2} (T, p) - g_{1} (T, p) = (v_{2} - v_{1}) (p - p_{0}) = 0$ . It vanishes at the center and has $\partial_{p} F = v_{2} - v_{1} \neq 0$ , so the $1 \times 1$ minor is invertible.

4.1F1F2A1step 3.1algebra∎

Here the ambient dimension is $d = 2 + P (C - 1) = 2$ and the equation count is $q = C (P - 1) = 1$ . Applying [F1] gives local equation-set dimension $d - q = 1$ , exactly $C - P + 2 = 1$ . This verifies the regular equation-set rank calculation only.

One component in two phases has one regular equation-set degree of freedom

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources