ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

Confined negative-capacity Kepler preparation

Physical scope

One energy and shell-radius instance of the finite regulated relative Kepler model under its declared Gibbs-volume convention and weak-field restriction; the canonical comparison remains a distinct preparation with unbounded momenta.

Example

Choose $m_{1}, m_{2} > 0$ , $A = G m_{1} m_{2}$ , reduced mass $μ = m_{1} m_{2} / (m_{1} + m_{2})$ , inner radius $r_{0} > 0$ , and outer radius $R > 32 r_{0}$ . Set $E = - A / (32 r_{0})$ and assume $2 A / (μ r_{0} c^{2}) \leq δ_{N R} ≪ 1$ . Use the two preparations and response conventions of the A-page comparison theorem.

Facts & Assumptions

Given: $R > 32 r_{0}$ , $E = - A / (32 r_{0})$ , $A, μ, r_{0} > 0$ , and the stated weak-field slow-motion condition.

[F1]

Under these hypotheses, the microcanonical Gibbs-volume response is strictly negative, and the separate canonical response for the same relative Hamiltonian is nonnegative. (Regulated gravitational preparations have different responses)

Verification

1.1givenF1algebra

The selected energy has $e = - E = A / (32 r_{0})$ , so $A / R < e < A / (16 r_{0})$ and $δ = e r_{0} / A = 1 / 32$ . The weak-field restriction is the condition required in [F1].

2.1F1step 1.1∎

Apply [F1] to this branch point. It gives $C_{G} < 0$ for the declared Gibbs-volume microcanonical preparation and $C_{K} \geq 0$ for the separate canonical preparation. These are model deductions about distinct finite ensembles, not measurements or evidence that either preparation is realized.

Depends on

Regulated gravitational preparations have different responses

Confined negative-capacity Kepler preparation

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources