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.

Regulated Kepler branch has negative volume response

Example

Choose relative parameters $μ > 0$ , $A > 0$ , $r_{0} > 0$ , and $R > 32 r_{0}$ . Set $e = A / (32 r_{0})$ , so $E = - e$ and $δ = e r_{0} / A = 1 / 32$ . This is an interior point of the finite regulated shell branch.

Facts & Assumptions

Given: The regulated relative shell, $R > 32 r_{0}$ , and the Gibbs-volume and canonical temperature conventions.

[F1]

On every regulated shell energy with $0 < δ \leq 1 / 16$ , the Gibbs-volume heat capacity is strictly negative. (Negative Gibbs-volume capacity in a finite regulated shell)

[F2]

For the same fixed relative Hamiltonian under a separate canonical preparation, the response is $C_{K} = k_{B} β^{2} Var (h) \geq 0$ . (Exact relative-shell canonical normalization and response)

Verification

1.1givenalgebra

Since $R > 32 r_{0}$ , $A / R < A / (32 r_{0}) = e$ ; also $e = A / (32 r_{0}) \leq A / (16 r_{0})$ . Thus the branch assumptions hold and $δ = 1 / 32 \leq 1 / 16$ .

2.1F1F2step 1.1∎

Applying [F1] gives $C_{G} < 0$ for the explicit Gibbs-volume entropy convention. Applying [F2] to the same relative Hamiltonian but its canonical preparation gives $C_{K} \geq 0$ . These are different preparations and response definitions, so the two signs do not contradict one another.

Regulated Kepler branch has negative volume response

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources