LemmaStatement: AI-adaptedProof: AI-generated

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.

The quantitative asymptotic end is uniformly timelike

Statement

For the AIK quantitative package, $g (T, T) \leq - 1 / 2$ on $Σ_{R_{1}}$ . In particular the stationary field is timelike throughout that end, independently of the horizon-smallness parameter.

Facts & Assumptions

[A1]

Work in ZFC, including the axiom of choice and its inherited countable and dependent choice uses. The Axiom of Choice

[F1]

The source end coordinates have $T = \partial_{0}$ , $∣ g_{00} + 1 - 2 M_{0} / r ∣ \leq A_{1} r^{- 2}$ and the chosen radius satisfies $R_{1} \geq 8 M_{0}$ and $R_{1} \geq 2 \sqrt{A_{1}}$ . Exact AIK global, horizon and quantitative hypotheses

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the stated conditional conclusion.

1.1F1givenalgebra

On the end the stationary field is the coordinate vector $\partial_{0}$ , so $g (T, T) = g_{00}$ . The zeroth-order term of the quantitative end bound gives $g_{00} \leq - 1 + 2 M_{0} / r + A_{1} / r^{2}$ .

2.1step 1.1F1algebra∎

For $r \geq R_{1}$ , the two positive terms are at most $1 / 4$ each by the chosen lower bounds on $R_{1}$ . Hence $g (T, T) \leq - 1 + 1 / 4 + 1 / 4 = - 1 / 2$ , as asserted. No horizon estimate enters this calculation.

Depends on

Used by · 1

The non-timelike region is confined to the local-rigidity collar

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · Section 1.2, equations (1.6)–(1.10), PDF pages 5–6