ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Entropy variables leave compact positive-state tubes near vacuum

Example

Take $R = 1$ , $γ = 2$ , $c_{v} = 1$ , and $s_{0} = 0$ in one dimension. For each integer $n \geq 1$ , let $U_{n} = (ρ_{n}, m_{n}, E_{n}) = (1 / n, 0, 1 / n)$ . Every state is admissible and has $p_{n} = 1 / n$ , $T_{n} = 1$ , and $s_{n} = \log n$ . Its density entropy variable is $v_{ρ, n} = 2 - \log n$ , which tends to $- \infty$ as the positive density and pressure tend to zero. The sequence approaches the excluded vacuum state $(0, 0, 0)$ , so no compact set $C ⋐ G$ contains all $U_{n}$ .

Verification

Given: The model and entropy-variable formulas with $R = 1$ , $γ = 2$ , $c_{v} = 1$ , $s_{0} = 0$ , and $U_{n} = (1 / n, 0, 1 / n)$ .

[F1] Ideal-gas compressible Navier–Stokes–Fourier model: $G = {ρ > 0, E > ∣ m ∣^{2} / (2 ρ)}$ , $p = (γ - 1) (E - ∣ m ∣^{2} / (2 ρ))$ , $T = p / (R ρ)$ , and $s = c_{v} \log (p / ρ^{γ}) + s_{0}$ .

[F2] Strict convexity and entropy variables for the ideal-gas Euler flux: $v_{ρ} = c_{v} γ - s - β ∣ u ∣^{2} / 2$ , $v_{E} = - β$ , and $β = 1 / T$ .

1.1F1givenalgebra

For $U_{n}$ , $ρ_{n} > 0$ , $ε_{n} = E_{n} = 1 / n > ∣ 0 ∣^{2} / (2 ρ_{n})$ , so $U_{n} \in G$ . The formulas [F1] give $p_{n} = 1 / n$ , $T_{n} = 1$ , and $s_{n} = \log ((1 / n) / (1 / n)^{2}) = \log n$ .

2.1F2step 1.1algebra

Here $u_{n} = 0$ and $β_{n} = 1$ . Substitution into [F2] gives $v_{ρ, n} = 2 - \log n$ and $v_{E, n} = - 1$ , hence $∣ v_{ρ, n} ∣ \to \infty$ .

3.1F1step 1.1step 2.1algebra∎

The states satisfy $U_{n} \to (0, 0, 0)$ , which is outside $G$ because its density is zero. Every compact subset of Euclidean state space is closed, so a compact $C ⋐ G$ containing every $U_{n}$ would also contain their limit, a contradiction.

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Sources

F. Ismail and P. L. Roe, Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks, Journal of Computational Physics 228 (2009), 5410–5436, complete paper (standard reference, not scraped)