ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck pass

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.

Prepared finite softened two-body ensemble

Physical scope

One explicit finite box and positive softening instance of the adopted effective Newtonian preparation; establishes a normalized model probability, not observed equilibration or a claim about a real gravitating gas.

Example

Take $B = (0, L)^{3}$ with $L > 0$ , $N = 2$ , equal mass $m > 0$ , Newton constant $G > 0$ , softening length $a > 0$ , action reference $h_{*} > 0$ , and $β > 0$ . Use the finite softened Newtonian preparation and its fixed Hamiltonian and measure.

Facts & Assumptions

Given: The cube has volume $L^{3} \in (0, \infty)$ , $N = 2$ , $a > 0$ , and all remaining physical parameters are positive as required by the adopted preparation.

[F1]

Under those hypotheses the finite regulated preparation has a positive finite partition function and normalized Gibbs density. (Constructed finite canonical gravitational equilibrium)

Verification

1.1givenF1

The box $B = (0, L)^{3}$ is bounded and has positive finite volume, and $N = 2 \geq 2$ with $a > 0$ ; therefore these data satisfy the hypotheses recorded in [F1].

2.1F1step 1.1∎

Apply [F1] to this instance. It gives $0 < Z_{2} (β) < \infty$ and $p_{β} = Z_{2}^{- 1} e^{- β H_{2, a}}$ as a normalized probability density relative to the stated phase reference. This is a constructed model ensemble and does not assert that the system dynamically reaches it.

Depends on

Constructed finite canonical gravitational equilibrium

Prepared finite softened two-body ensemble

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources