ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

Extremal RN smooth horizon with zero kappa

Example

Let $M > 0$ and $q_{e}, q_{m} \in R$ satisfy $q_{e}^{2} + q_{m}^{2} = M^{2}$ . Then the selected ingoing RN model has $f (r) = (1 - M / r)^{2}$ , a double positive root $r_{H} = M$ , and surface gravity $κ_{H} = 0$ . Its ingoing metric remains nondegenerate at $r = M$ . The simple-root bifurcation chart constructed in Regular RN crossings and local bifurcation extension requires $κ > 0$ and does not apply here. No separate extremal bifurcation chart or physical realization is asserted.

Facts & Assumptions

Given: The selected ingoing RN data, $M > 0$ and $q_{e}^{2} + q_{m}^{2} = M^{2}$ .

[F1]

RN roots, Killing horizons and surface gravity: at the double positive root, $r_{H} = M$ and $κ_{H} = 0$ under these parameter assumptions.

[F2]

Regular RN crossings and local bifurcation extension: the $v, r$ block is nondegenerate at every positive root; its additional bifurcation chart is proved only for a simple root with $κ > 0$ . Its field-equation continuation remains conditional on the current RN verification draft.

[F3]

Selected ingoing RN extension and time orientation: the selected metric is $- f d v^{2} + 2 d v d r + r^{2} γ$ on $r > 0$ .

[A1]

All parameters and $r$ have length units; $κ$ has inverse-length units. The mathematical example concerns the specified metric only.

Verification

1.1F1A1algebra

Substituting $q_{e}^{2} + q_{m}^{2} = M^{2}$ gives $f (r) = 1 - \frac{2 M}{r} + \frac{M^{2}}{r^{2}} = (1 - M / r)^{2} .$ Thus $f (M) = f^{'} (M) = 0$ , so the positive root is double. By [F1], its surface gravity is $f^{'} (M) / 2 = 0$ .

2.1F2F3A1step 1.1algebra

The $v, r$ block in [F3] has determinant $- 1$ for every $f$ , and at $r = M$ the sphere metric $M^{2} γ$ is positive definite. Hence the metric is nondegenerate and Lorentzian at the double root. The extra chart in [F2] assumes a simple root and $κ > 0$ , both of which fail by step 1.1; that construction therefore supplies no extremal bifurcation chart.

3.1F1F2step 1.1step 2.1∎

Steps 1.1 and 2.1 establish the double-root, zero-surface-gravity case and the regular ingoing metric while delimiting the simple-root construction. No maximal extension or physical existence conclusion follows.

Remarks

This is a conditional model calculation; it does not assert that any physical system realizes the extremal parameters.

Extremal RN smooth horizon with zero kappa

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources