Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-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.

Constructed regular-center local static TOV branch

Physical scope

Local branch around a regular center for a supplied smooth positive EOS and compact central-pressure range in the adopted classical static isotropic Lambda=0 model; no global surface, global horizon avoidance, realistic EOS, empirical claim, or dynamical stability.

Statement

Assume the adopted model Static spherical Einstein-fluid model with a supplied equation of state with its supplied $C^{\infty}$ positive EOS $E : I_{P} \to (0, \infty)$ on an open pressure interval $I_{P} \subset (0, \infty)$ . Let $P_{c} \subset I_{P}$ be a nonempty compact interval, let $δ > 0$ be such that its closed $δ$ -neighbourhood lies in $I_{P}$ , and choose a central pressure $P_{0} \in P_{c}$ and a dimensionless central lapse $e^{ϕ_{c}} > 0$ . Under the exact mathematical convention $Z F + {A C}_{ω}$ of Regular-center TOV local existence and dependence, put $K = 4 π G / c^{4}$ . There is a common $R_{0} > 0$ and, for each $P_{0} \in P_{c}$ , smooth even profiles $P, v$ on $[- R_{0}, R_{0}]$ with $P (r) \in I_{P}$ for every $∣ r ∣ \leq R_{0}$ and satisfying $v (r) = K \int_{0}^{1} s^{2} E (P (r s)) d s, P (r) = P_{0} - \int_{0}^{r} t \frac{(E (P (t)) + P (t)) (v (t) + K P (t))}{1 - 2 t^{2} v (t)} d t,$ with $P (0) = P_{0}$ , $v (0) = K E (P_{0}) / 3$ , $b (r) : = 1 - 2 r^{2} v (r) \geq 1 / 2$ , and $P^{'} (0) = v^{'} (0) = 0$ . Defining $M (r) = \frac{c^{2}}{G} r^{3} v (r), ϕ (r) = ϕ_{c} + \int_{0}^{r} t \frac{v (t) + K P (t)}{1 - 2 t^{2} v (t)} d t,$ gives a positive-lapse $C^{2}$ metric center and a local solution of the adopted Einstein-fluid equations. On $r > 0$ its equations are $M^{'} = \frac{4 π r^{2} E (P)}{c^{2}}, ϕ^{'} = \frac{G M / c^{2} + 4 π G r^{3} P / c^{4}}{r^{2} b}, P^{'} = - \frac{G (E (P) + P) (M + 4 π r^{3} P / c^{2})}{c^{2} r^{2} b} .$ No conclusion is made that the branch reaches a finite zero-pressure surface, remains horizon-free globally, or is dynamically stable.

Facts & Assumptions

Given: The adopted static Einstein-fluid model, a smooth positive supplied EOS, a compact central-pressure range with the stated margin, $ϕ_{c} \in R$ , and the exact $Z F + {A C}_{ω}$ convention of the mathematical suppliers.

[F1]

Static spherical Einstein-fluid model with a supplied equation of state adopts the $Λ = 0$ field equation on each open fluid interval $J$ , where its pressure map is $P \in C^{\infty} (J, I_{P})$ , the restriction of the earlier pressure component $p$ , and its energy density is $E \circ P$ ; $E$ is a supplied smooth positive function, not a derived EOS.

[F2]

Regular-center TOV local existence and dependence: for positive smooth $E$ on $I_{P}$ , compact $P_{c}$ with a closed pressure margin, and $ϕ_{c} \in R$ , the even Volterra profiles displayed in the Statement exist smoothly on a common interval and obey $∣ P (r) - P_{0} ∣ \leq δ$ , hence $P (r) \in I_{P}$ ; they satisfy $b \geq 1 / 2$ and give a $C^{2}$ Cartesian center and local continuity in $P_{0}$ . The theorem assumes $Z F + {A C}_{ω}$ .

[F3]

Complete spherical Einstein and conservation reduction: on every smooth $r > 0$ interval with $b > 0$ , the mass, lapse, and pressure equations are equivalent to all mixed Einstein equations under the stated $(-, +, +, +)$ curvature convention.

[A1]

The field equation and EOS in [F1] are adopted model premises, not deductions about nature.

Proof

1.1givenA1F1F2

The supplied $E$ , compact interval $P_{c}$ , pressure margin $δ$ , and central lapse meet [F2] exactly. Its Volterra construction gives the even smooth $P, v$ , the lower bound $b \geq 1 / 2$ , and the displayed integral formula for $ϕ$ ; in particular the center is regular in the stated $C^{2}$ Cartesian sense.

2.1A1F1F2F3step 1.1algebra∎

The constructed profiles satisfy the mass, pressure, and lapse equations in [F3] on $r > 0$ , with $ε (r) = E (P (r))$ and $b > 0$ . The converse direction of [F3] therefore gives every mixed Einstein component of the adopted model on the punctured neighborhood. In Cartesian coordinates the static unit vector $u = e^{- ϕ} \partial_{x^{0}}$ and metric are continuous, and the isotropic stress has covariant form $T_{a b} = (ε + P) u_{a} u_{b} + P g_{a b}$ , so its components extend continuously to the center. A $C^{2}$ metric has a continuous Einstein tensor; equality on the punctured neighborhood therefore extends to the center.

Constructed regular-center local static TOV branch

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources