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.

Black and white exclusion is an ambient reachability condition

Statement

In a time-oriented spacetime $M$ , fix a union of stationary end regions $M_{e x t}$ and any subset $U \subset M$ . Define $B = M ∖ I_{M}^{-} (M_{e x t})$ and $W = M ∖ I_{M}^{+} (M_{e x t})$ . Then $(B \cup W) \cap U = \emptyset$ if and only if $U \subset I_{M}^{-} (M_{e x t}) \cap I_{M}^{+} (M_{e x t})$ . For any subset $Σ \subset M$ , define $D (Σ)$ here to be the set of points through which every inextendible timelike curve in $M$ meets $Σ$ . The equivalence applies in particular to $U = int D (Σ)$ with this explicit local convention. The chronological relations are computed in $M$ ; changing the ambient spacetime requires recomputing them.

Facts & Assumptions

[F1]

Chronological past and future use future timelike curves in the specified ambient manifold. Causal exterior, chronological boundaries and Cauchy data

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the stated construction and identities.

1.1F1

The equality $(B \cup W) \cap U = \emptyset$ means simultaneously that $U \cap B$ and $U \cap W$ are empty. Taking complements in $M$ gives $U \subset I_{M}^{-} (M_{e x t})$ and $U \subset I_{M}^{+} (M_{e x t})$ . Their conjunction is precisely the stated inclusion.

2.1F1step 1.1∎

Conversely the inclusion excludes every point of $U$ from each complement, so it excludes their union. This set argument holds for any $U$ , in particular the interior of the domain of dependence defined above; it neither identifies that domain with a desired exterior nor changes timelike-curve domains of dependence to a different convention.

Depends on

Causal exterior, chronological boundaries and Cauchy data

Used by · 2

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.