TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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.

Exact spherical horizon charge

Statement

Fix $m > 0$ and real $q_{e}, q_{m}$ with $q_{e}^{2} + q_{m}^{2} = m^{2}$ , and take the ingoing extremal RN metric and coordinates specified by the spherical test-scalar definition. Let $ψ$ be any admissible neutral spherical $C^{3}$ solution of $□_{g} ψ = 0$ , set $ϕ = r ψ$ , and define its spherical horizon charge by

$H_{ψ} (v) : = \partial_{r} ϕ (v, m) .$

Then $H_{ψ}$ is constant for all $v \in R$ . If this constant is nonzero, the regular ingoing-chart transverse derivative $\partial_{r} ϕ (v, m)$ is nonzero and does not tend to zero along the horizon. If $[ψ]$ is the scalar-field unit, then $[H_{ψ}] = [ψ]$ . This is a local fixed-background statement and asserts no global decay or nonlinear instability.

Statement references

Spherical scalar waves in regular extremal RN coordinates

The derivative of the metric density and the chart formula for the divergence of a vector field

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: $m > 0$ , $q_{e}^{2} + q_{m}^{2} = m^{2}$ , ${A C}_{ω}$ , and an admissible real spherical $C^{3}$ neutral solution $ψ$ of $□_{g} ψ = 0$ on the selected ingoing RN model.

[F1]

Spherical scalar waves in regular extremal RN coordinates: the background is $g = - D d v^{2} + 2 d v d r + r^{2} γ_{S^{2}}$ with $D = (1 - m / r)^{2}$ and fixed Maxwell field $F_{i n} = - q_{e} r^{- 2} d v \land d r + q_{m} ω_{S^{2}}$ . The vector field $ξ = \partial_{v}$ generates the null level $r = m$ with zero signed surface gravity. The reduced $C^{3}$ profile $ψ (v, r)$ lifts to $\tilde{ψ} (v, r, ω) = ψ (v, r)$ , $ϕ = r ψ$ , and the wave operator on this lift is the trace of its $C^{1}$ Hessian, equivalently $w^{- 1} \partial_{a} (w g^{a b} \partial_{b} \tilde{ψ})$ ; it is $C^{1}$ . All radial derivatives are ingoing-chart derivatives.

[F2]

The derivative of the metric density and the chart formula for the divergence of a vector field: for a smooth vector field $J$ , $\nabla_{a} J^{a} = w^{- 1} \partial_{a} (w J^{a})$ , where $w = \sqrt{∣ \det (g_{a b}) ∣}$ . Its coordinate proof uses only first derivatives of $J$ and the product rule, so the same formula holds for $C^{1}$ vector fields. Since $ψ$ is $C^{3}$ and $g$ is smooth, $J^{a} = g^{a b} \partial_{b} \tilde{ψ}$ is $C^{2}$ and the extension applies.

[F3]

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: the metric-selected connection is metric compatible and torsion free; in particular $\nabla_{a} g^{b c} = 0$ and its tensor derivatives have the stated coordinate formula.

[F4]

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant: a continuous real function on an order-convex interval whose derivative vanishes at every interior point is constant on that interval.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the tangent-bundle smooth structure used by [F1] and [F3] assumes ${A C}_{ω}$ . This is the only choice assumption inherited here; no full axiom of choice is used.

Proof

1.1givenF1F2F3A1algebra

In a local sphere chart with angular metric components $γ_{A B}$ , the ingoing $(v, r)$ metric block and its inverse are $(\begin{array}{cc} - D & 1 \\ 1 & 0 \end{array}), (\begin{array}{cc} 0 & 1 \\ 1 & D \end{array}) .$ The first block has determinant $- 1$ , so the full metric density is $w = r^{2} \sqrt{\det (γ_{A B})}$ . Put $J^{a} : = g^{a b} \partial_{b} \tilde{ψ}$ . The spherical lift gives $J^{v} = ψ_{r}$ , $J^{r} = ψ_{v} + D ψ_{r}$ , and $J^{A} = 0$ ; this vector field is $C^{2}$ , so [F2] applies. By [F1], [F3] and metric compatibility, $□_{g} \tilde{ψ} = \nabla_{a} J^{a}$ . Applying the density formula and cancelling the angular density, which is independent of $v, r$ , gives $0 = □_{g} \tilde{ψ} = \frac{1}{r^{2}} [\partial_{v} (r^{2} ψ_{r}) + \partial_{r} ⁣ (r^{2} (ψ_{v} + D ψ_{r}))] .$ Expanding the derivatives and using equality of the continuous mixed partials yields $2 r^{2} ψ_{v r} + 2 r ψ_{v} + \partial_{r} (r^{2} D ψ_{r}) = 0.$

2.1step 1.1F1algebra

Since $ϕ = r ψ$ , one has $2 r^{2} ψ_{v r} + 2 r ψ_{v} = 2 r ϕ_{v r}$ and $r^{2} D ψ_{r} = D (r ϕ_{r} - ϕ)$ . Differentiating the latter expression gives $\partial_{r} (r^{2} D ψ_{r}) = r D ϕ_{r r} + r D^{'} ϕ_{r} - D^{'} ϕ$ . Divide the equation of step 1.1 by $r > 0$ to obtain $2 ϕ_{v r} + D ϕ_{r r} + D^{'} ϕ_{r} - \frac{D^{'}}{r} ϕ = 0.$

3.1step 2.1F1algebra

From $D = (1 - m / r)^{2}$ it follows by direct differentiation that $D (m) = D^{'} (m) = 0$ . Substituting $r = m$ into step 2.1 leaves $2 ϕ_{v r} (v, m) = 0$ . The assumed $C^{3}$ regularity makes $H_{ψ} (v) = ϕ_{r} (v, m)$ continuous and differentiable with $H_{ψ}^{'} (v) = ϕ_{r v} (v, m) = ϕ_{v r} (v, m) = 0$ .

4.1F1F4step 3.1algebra∎

The real line is an order-convex interval, so [F4] applied to $H_{ψ}$ and step 3.1 shows that $H_{ψ} (v) = H_{ψ} (v_{0})$ for every pair $v, v_{0} \in R$ . If this common value is nonzero, the defining equality $\partial_{r} ϕ (v, m) = H_{ψ} (v)$ shows that the ingoing transverse derivative is the same nonzero number for every $v$ and therefore cannot tend to zero.

Remarks

The charge is the derivative of $ϕ = r ψ$ in the regular ingoing chart. It is not the derivative of $ψ$ alone and is not computed in a singular static chart. In Aretakis Part I's convention $H_{0} = \partial_{r} ψ + ψ / m$ , this normalization is $H_{ψ} = m H_{0}$ , since $\partial_{r} (r ψ) ∣_{r = m} = ψ + m \partial_{r} ψ$ . The result is proved directly from the local wave equation; no decay estimate or initial-data theorem is assumed.

Exact spherical horizon charge

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources