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.

Finite softened two-body box

Example

Let $B = [0, L]^{3}$ with $L > 0$ , choose $N = 2$ , mass $m > 0$ , regulator $a > 0$ , action reference $h_{*} > 0$ , and inverse temperature $β > 0$ . Use the softened Hamiltonian $H (q_{1}, q_{2}, p_{1}, p_{2}) = \frac{∣ p_{1} ∣^{2} + ∣ p_{2} ∣^{2}}{2 m} - \frac{G m^{2}}{\sqrt{∣ q_{1} - q_{2} ∣^{2} + a^{2}}}$ on $B^{2} \times R^{6}$ with measure $d^{3} q_{1} d^{3} q_{2} d^{3} p_{1} d^{3} p_{2} / (2! h_{*}^{6})$ .

Facts & Assumptions

Given: The finite softened model, a positive-volume box, $m, G, a, β, h_{*} > 0$ .

[F1]

Every finite softened model on a positive finite-volume box has a finite positive Gibbs partition function and a normalized density for each $β > 0$ . (Finite gravitational Gibbs normalization and differentiable moments)

Verification

1.1givenF1algebra

For every $(q_{1}, q_{2}) \in B^{2}$ , the pair attraction is between $- G m^{2} / a$ and $0$ , so the Hamiltonian has the exact form covered by [F1]. The box has volume $L^{3} > 0$ and finite, so all hypotheses apply.

2.1F1step 1.1∎

By [F1], $0 < Z_{2} (β) < \infty$ and $p_{β} = Z_{2}^{- 1} e^{- β H}$ is a probability density relative to the stated dimensionless phase measure. This verifies the finite softened two-body preparation for every $β > 0$ ; it makes no claim about the distinct point model or dynamical equilibration.

Depends on

Finite gravitational Gibbs normalization and differentiable moments

Finite softened two-body box

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources