ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

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.

Positive regular constant-density branch

Example

Fix $R > 0$ and choose $ρ_{0} = \frac{3 c^{2}}{16 π G R^{2}}, M_{R} = \frac{4 π ρ_{0} R^{3}}{3}, u = \frac{2 G M_{R}}{c^{2} R} = \frac{1}{2} .$ With $ϵ_{0} = ρ_{0} c^{2}$ and $a = 1 / \sqrt{2}$ , the exact interior profile is $b (r) = 1 - \frac{r^{2}}{2 R^{2}}, p (r) = ϵ_{0} \frac{\sqrt{1 - r^{2} / (2 R^{2})} - 1 / \sqrt{2}}{3 / \sqrt{2} - \sqrt{1 - r^{2} / (2 R^{2})}}, e^{ϕ (r)} = \frac{3 / \sqrt{2} - \sqrt{1 - r^{2} / (2 R^{2})}}{2} .$

Facts & Assumptions

Given: The chosen $R, ρ_{0}, M_{R}, u$ and the exact-solution theorem Exact regular constant-density solution with verified matching, under $Z F + {A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

For every $ρ_{0}, R > 0$ with $u = 2 G M_{R} / (c^{2} R) \in (0, 8 / 9)$ , Exact regular constant-density solution with verified matching verifies the interior and vacuum equations, regular center, matching, and central pressure formula $ϵ_{0} (1 - a) / (3 a - 1)$ .

[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

1.1givenA1F1algebra

Substitution of the chosen $ρ_{0}$ gives $M_{R} = c^{2} R / (4 G)$ and hence $u = 1 / 2 \in (0, 8 / 9)$ . Therefore [F1] applies to the displayed functions.

2.1A1F1step 1.1algebra∎

The central pressure ratio is $\frac{p (0)}{ϵ_{0}} = \frac{1 - 1 / \sqrt{2}}{3 / \sqrt{2} - 1} = \frac{2 \sqrt{2} - 1}{7} > 0,$ and the central lapse is $(3 / \sqrt{2} - 1) / 2 > 0$ ; the surface pressure is $p (R) = 0$ and surface lapse is $1 / \sqrt{2}$ . The theorem gives the regular Cartesian center and no-layer matching to the vacuum exterior.

Remarks

The profile is an exact solution of the conditional equation system. This mathematical example is not an observed star and supplies no causal transport or stability law.

Positive regular constant-density branch

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources