ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Half-compactness constant-density Einstein star

Physical scope

The explicit constant-density family at u=1/2, with its adopted vacuum exterior; not an observed star, realistic EOS, or generic compactness result.

Example

Specialize the constant-density family of Verified constant-density static star and vacuum exterior to $u = 1 / 2$ . Then $a = \frac{1}{\sqrt{2}}, \frac{P (0)}{ε_{0}} = \frac{\sqrt{2} - 1}{3 - \sqrt{2}} \in (0, 1), e^{ϕ (0)} = \frac{3 / \sqrt{2} - 1}{2} > 0.$ At the surface, $b (R) = 1 / 2$ and $e^{ϕ (R)} = 1 / \sqrt{2}$ . The prescribed vacuum exterior has $M = M_{R}$ , $b (r) = 1 - R / (2 r)$ and $e^{ϕ (r)} = \sqrt{b (r)}$ for $r \geq R$ .

Facts & Assumptions

Given: The exact static star family and its equations from Verified constant-density static star and vacuum exterior, with the specialization $u = 1 / 2$ .

[F1]

Verified constant-density static star and vacuum exterior gives $a = \sqrt{1 - u}$ , $B (r) = \sqrt{1 - u r^{2} / R^{2}}$ , $P (0) / ε_{0} = (1 - a) / (3 a - 1)$ , $e^{ϕ (0)} = (3 a - 1) / 2$ , and the vacuum exterior $b (r) = 1 - u R / r$ , $e^{ϕ} = \sqrt{b}$ .

Verification

1.1givenF1algebra

Substitution of $u = 1 / 2$ gives $a = 1 / \sqrt{2}$ and hence $P (0) / ε_{0} = (\sqrt{2} - 1) / (3 - \sqrt{2})$ . Both numerator and denominator are positive, and $\sqrt{2} - 1 < 3 - \sqrt{2}$ because $\sqrt{2} < 2$ ; thus the ratio lies strictly between zero and one. Also $e^{ϕ (0)} = (3 / \sqrt{2} - 1) / 2 > 0$ , $b (R) = a^{2} = 1 / 2$ , and $e^{ϕ (R)} = a = 1 / \sqrt{2}$ .

2.1F1step 1.1∎

With $u = 1 / 2$ , the exterior formula in [F1] becomes $b (r) = 1 - R / (2 r)$ ; it is positive for $r \geq R$ and has $e^{ϕ} = \sqrt{b}$ . The theorem supplies the matching and field-equation verification for this family, so this specialization is a finite, positive-pressure example within the adopted model.

Depends on

Verified constant-density static star and vacuum exterior

Dependency tree · 2 levels

Sources

MIT 8.962, General Relativity, Spring 2020 Lecture Guide, Lecture 21, PDF pp. 8–10