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.

Two-sided far-tail access preserves the ambient causal exterior

Statement

Assume the Axiom of Choice. For a specified end region $U \subset M$ , define its domain of outer communications (DOC) in the fixed ambient spacetime $M$ by $D_{M} (U) : = I_{M}^{-} (U) \cap I_{M}^{+} (U)$ . Let the original ambient chronological exterior be $E = D_{M} (U_{0}) = I_{M}^{-} (U_{0}) \cap I_{M}^{+} (U_{0})$ and suppose that, with the metric and time orientation induced from $M$ , $E$ satisfies all the hypotheses of the maximal stationary future-access lemma. For a sufficiently far stationary tail $U_{f} \subset U_{0} \cap E$ , assume this is the tail identified with $R \times {r > R_{f}}$ in that product. Then $I_{M}^{-} (U_{f}) \cap I_{M}^{+} (U_{f}) = E$ . In particular $D_{M} (U_{f}) = D_{M} (U_{0})$ : the original DOC is preserved when its designated end is truncated to any such sufficiently far stationary tail. No assertion about equality of each individual past or future set on the whole ambient manifold is included.

Statement references

A maximal stationary exterior has future access to its far tail

Past access to the far tail by exact time reversal

Causal curves, chronological and causal reachability, and inextendibility

The Axiom of Choice

Facts & Assumptions

[A1]

The Axiom of Choice is assumed to meet the foundational hypothesis of both far-tail access suppliers. The Axiom of Choice

[F1]

Under [A1] and all the future-access hypotheses in the Statement, every $E$ point reaches $U_{f}$ along a future timelike curve contained in $E$ , and is reached from $U_{f}$ along one. A maximal stationary exterior has future access to its far tail Past access to the far tail by exact time reversal

[F2]

Chronological reachability is existential over timelike curves, so inclusion of domains or endpoint sets gives the corresponding inclusion of chronological sets. Causal curves, chronological and causal reachability, and inextendibility

Proof

**Given:**The explicit hypotheses of the Statement.

Goal: Establish its exact chronological-access conclusion.

1.1givenA1F1F2

The Axiom of Choice in [A1] and the full future-access hypotheses in the Statement meet the hypotheses of both access lemmas in [F1]. For $p \in E$ , the two curves of [F1] are also timelike curves in $M$ . Consequently $p \in I_{M}^{-} (U_{f}) \cap I_{M}^{+} (U_{f})$ , proving inclusion of the original exterior in the new one. The fact that the paths remain in $E$ is part of [F1], not an inference from the old definition alone.

2.1givenF2step 1.1∎

Conversely $U_{f} \subset U_{0}$ implies $I_{M}^{-} (U_{f}) \subset I_{M}^{-} (U_{0})$ and $I_{M}^{+} (U_{f}) \subset I_{M}^{+} (U_{0})$ . Intersecting and using the definition of $E$ gives the reverse inclusion. By the definition of $D_{M}$ , this also gives $D_{M} (U_{f}) = E = D_{M} (U_{0})$ , proving preservation of the DOC. Separate one-sided ambient identities remain unasserted.

Depends on

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