LemmaStatement: AI-generatedProof: 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.

Complete stationary saturation of the end

Statement

For the complete stationary action $ϕ_{u}$ of Exact asymptotically stationary end contract, define the saturated end by $M_{e n d} = ⋃_{u \in R} ϕ_{u} (Σ_{1})$ . Then $ϕ_{v} (M_{e n d}) = M_{e n d}$ for every $v \in R$ , and $T$ is timelike everywhere on $M_{e n d}$ . The saturation uses $T$ and need not be defined using a different horizon generator.

Facts & Assumptions

[F1]

The end contract supplies a complete group of isometries generated by $T$ , with $g (T, T) < - C^{- 1}$ on $Σ_{1}$ . Exact asymptotically stationary end contract

Proof

Given: The complete action and saturated end in the Statement.

Goal: Establish saturation invariance and timelikeness on the saturation.

1.1F1

The group law gives $ϕ_{v} (M_{e n d}) = ⋃_{u \in R} ϕ_{v + u} (Σ_{1}) = M_{e n d}$ , since translation by $v$ is a bijection of $R$ . Completeness supplies every parameter used here.

2.1F1step 1.1∎

Differentiating $ϕ_{v} (ϕ_{u} (p)) = ϕ_{v + u} (p)$ at $u = 0$ gives $(d ϕ_{v})_{p} T_{p} = T_{ϕ_{v} (p)}$ . Since $ϕ_{v}$ is an isometry, $g (T, T) (ϕ_{v} (p)) = g (T, T) (p)$ . For $p \in Σ_{1}$ this is negative, proving timelikeness on its full saturation. Both assertions concern the specified stationary action only.

Depends on

Exact asymptotically stationary end contract

Used by · 2

Dependency tree · 2 levels

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