ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge 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.

Local regular-center central-pressure family

Example

Let $ϵ_{0} > 0$ have units $J m^{- 3}$ , let $λ \geq 0$ be dimensionless, and prescribe the smooth positive state relation $ϵ (p) = ϵ_{0} + λ p$ for $p \in I_{p} = (0, \infty)$ . Fix $0 < p_{1} < p_{2}$ and let the central pressure range be $P_{c} = [p_{1}, p_{2}]$ . This is a mathematical equation-of-state example, not an observational stellar model.

Facts & Assumptions

Given: The state relation, the compact interval $P_{c}$ , a fixed central lapse value $ϕ_{c}$ , and $Z F + {A C}_{ω}$ inherited from The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

For every smooth positive $ϵ (p)$ on an open positive pressure interval and compact central pressure interval contained in it with a positive margin, Regular-center TOV local existence and dependence supplies a common $R_{0} > 0$ , the unique integral-equation profiles in the stated pressure ball, a $C^{2}$ Cartesian center, continuous dependence on $p_{c}$ , and continuation only within a compact subset where $r > 0$ , $p \in I_{p}$ , and $b > 0$ .

[A1]

The geometric supplier chain uses exactly ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ), with no full Axiom of Choice.

Verification

1.1givenA1algebra

The map $ϵ (p) = ϵ_{0} + λ p$ is $C^{\infty}$ and positive on $I_{p} = (0, \infty)$ . Since $p_{1} > 0$ , any $0 < δ < p_{1}$ gives the required closed pressure margin around $P_{c}$ inside $I_{p}$ .

2.1A1F1step 1.1algebra∎

Apply [F1] under [A1] with this $ϵ$ and $P_{c}$ . There is one radius $R_{0} > 0$ valid for every $p_{c} \in [p_{1}, p_{2}]$ , and the corresponding solutions depend continuously on $p_{c}$ in the uniform norm; they satisfy $v (0) = K (ϵ_{0} + λ p_{c}) / 3$ , $p^{'} (0) = v^{'} (0) = 0$ , and have positive lapse and $b \geq 1 / 2$ .

Remarks

This is local existence and parameter dependence for the stated mathematical system. No surface at $p = 0$ , global compactness bound, empirical equation of state, or dynamical stability conclusion follows.

Local regular-center central-pressure family

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources