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.

Chronological equivariance and the no-white-hole condition

Statement

Use the chronological relations of Causal exterior, chronological boundaries and Cauchy data. If a time-orientation-preserving isometry $ψ$ satisfies $ψ (E) = E$ , then $ψ (I^{\pm} (E)) = I^{\pm} (E)$ and $ψ (\partial I^{\pm} (E)) = \partial I^{\pm} (E)$ . In particular this applies to every $ϕ_{v}$ and $E = M_{e n d}$ of Complete stationary saturation of the end. Thus $B = M ∖ I^{-} (M_{e n d})$ , $W = M ∖ I^{+} (M_{e n d})$ and $N = \partial I^{-} (M_{e n d})$ are stationary-invariant, and $W = \emptyset$ is exactly $M = I^{+} (M_{e n d})$ . Chronological sets here are defined by the same curve formulas for any subset $E$ , without requiring $E$ open.

Facts & Assumptions

[F1]

Chronological futures and pasts are defined by existence of future timelike curves, and boundaries are closure minus interior. Causal exterior, chronological boundaries and Cauchy data

[F2]

Every $ϕ_{v}$ is an isometry and the saturated end is invariant under it. Complete stationary saturation of the end

Proof

Given: The action, set and causal conventions in the Statement.

Goal: Prove the chronological and boundary invariance and the no-white-hole identity.

1.1F1

A time-orientation-preserving isometry maps future timelike curves to future timelike curves. Applying it to a curve from $E$ to $p$ gives $ψ (I^{+} (E)) \subset I^{+} (E)$ ; applying its inverse gives the reverse inclusion. The identical argument with the curve direction reversed proves the past identity. As a diffeomorphism, $ψ$ maps open sets to open sets, and it and its inverse preserve limits, so it preserves interiors and closures and hence the displayed boundaries.

2.1F1F2step 1.1∎

The connected one-parameter action preserves time orientation: along $v \mapsto (d ϕ_{v})_{p} V$ , for a fixed future timelike $V$ , timelikeness is preserved and the sign relative to a continuous future time-orientation field cannot change without becoming zero. At $v = 0$ the vector is future. Apply step 1.1 and [F2]. Complements are preserved by a bijection, and $M ∖ I^{+} (M_{e n d}) = \emptyset$ is equivalent to $M = I^{+} (M_{e n d})$ by elementary set complementation. This proves all assertions.

Depends on

Used by · 1

Precise Rácz–Wald global collapse class

Dependency tree · 2 levels

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