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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.

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Charged photon trapping and extremal scaling — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Subextremal RN surface gravity has no uniform positive lower bound

Statement refuted

Fix a mass length $m > 0$ . For each length parameter $q_{e} \in R$ with $0 < q_{e}^{2} < m^{2}$ , take the electric RN family with mass parameter $M = m$ and magnetic parameter $q_{m} = 0$ , and define $Q^{2} : = q_{e}^{2}$ , $r_{\pm} : = m \pm \sqrt{m^{2} - Q^{2}}$ , and $κ_{+} (q_{e}) : = (r_{+} - r_{-}) / (2 r_{+}^{2})$ . The false claim is that there exists a constant $c > 0$ , with units of inverse length, such that $κ_{+} (q_{e}) \geq c$ for every such $q_{e}$ .

Statement references

RN roots, Killing horizons and surface gravity

Facts & Assumptions

Given: A fixed $m > 0$ and the subextremal electric RN family with $q_{m} = 0$ and $0 < q_{e}^{2} < m^{2}$ .

[F1]

Dyonic Reissner-Nordström metric and field domains: $Q^{2} = q_{e}^{2} + q_{m}^{2}$ and the static RN metric uses $f (r) = 1 - 2 m / r + Q^{2} / r^{2}$ .

[F2]

RN roots, Killing horizons and surface gravity: when $m > 0$ and $Q^{2} > 0$ , the outer surface gravity is $κ_{+} = (r_{+} - r_{-}) / (2 r_{+}^{2})$ with $r_{\pm} = m \pm \sqrt{m^{2} - Q^{2}}$ .

[F3]

Explicit rational and real power regimes: for $a > 0$ , $\sqrt{a}$ is the unique positive real solution of $s^{2} = a$ .

[F4]

Dedekind-complete ordered real field: $R$ is a complete ordered field.

[F5]

For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ : in any complete ordered field, for every $ϵ > 0$ there is a natural $N \geq 1$ with $1 / N < ϵ$ .

[A1]

The fixed mass parameter satisfies $m > 0$ ; all mass and charge parameters have length units and $κ_{+}$ has inverse-length units.

Counterexample

1.1givenA1F1F2F3algebra

For each integer $n \geq 2$ , choose $q_{e, n} = m \sqrt{1 - n^{- 2}}$ and $q_{m, n} = 0$ . By [F3], $Q_{n}^{2} = q_{e, n}^{2} = m^{2} (1 - n^{- 2})$ , so $0 < Q_{n}^{2} < m^{2}$ and $\sqrt{m^{2} - Q_{n}^{2}} = m / n$ . Applying [F2] gives $r_{+, n} = m (1 + 1 / n)$ , $r_{-, n} = m (1 - 1 / n)$ , and $κ_{+, n} = \frac{r_{+, n} - r_{-, n}}{2 r_{+, n}^{2}} = \frac{n}{m (n + 1)^{2}}, 0 < κ_{+, n} < \frac{1}{m n} .$

2.1givenA1F4F5step 1.1algebra∎

Let $c > 0$ be any proposed inverse-length lower bound. Since $m c > 0$ , [F4]–[F5] allow a natural $N \geq 1$ with $1 / N < m c$ . For every integer $n \geq 2 N$ , $1 / n \leq 1 / (2 N) < m c$ , so step 1.1 gives $0 < κ_{+, n} < 1 / (m n) < c$ . Thus the sequence $κ_{+, n}$ tends to zero through subextremal RN data, contradicting any positive uniform lower bound. Every member of the sequence remains subextremal; only its limit reaches $Q^{2} = m^{2}$ .

Remarks

The counterexample concerns the unscaled outer surface gravity as the charge approaches extremality at fixed mass. It makes no claim about the scaled throat coordinates or about uniform estimates for waves.

Sources

Max Weissenbacher, Decay and non-decay for the massless Vlasov equation on subextremal and extremal Reissner–Nordström black holes, Arch. Rational Mech. Anal. 248, 118 (2024)