LemmaStatement: AI-generatedProof: AI-generated

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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.

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First contact lies on the chronological and causal future boundary

Statement

Work in ZFC. At the earliest cylinder time, every contact point $p_{R}$ lies in $\partial I^{+} (A) = \partial J^{+} (A)$ and satisfies $τ (p_{R}) > 0$ for the supplied temporal function $τ$ . No point of the whole cut $C_{t_{R}}$ belongs to $I^{+}$ (A). These statements concern causal relations in the open exterior and do not assume a global event-horizon identity for the transformed metric.

Facts & Assumptions

[F1]

Use the full data and hypotheses of the compact-contact lemma in the same exterior $E$ : its compact source $A \subset τ^{- 1} (0)$ , cuts $C_{t} = Φ ({t} \times C_{0})$ , continuous parameter map $Φ$ , and finite earliest causal-contact time $t_{R}$ . Its conclusion supplies $C_{t_{R}} \cap J^{+} (A) \neq \emptyset$ and $τ (p) > 0$ for every causal contact point $p$ with the cylinder. This positivity concerns the temporal-function value; no sign for the cut parameter $t_{R}$ is supplied. Compactness gives an earliest translated-cylinder causal contact

[F2]

Chronological futures are open. Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets

[F3]

A causal segment followed or preceded by a timelike segment gives a timelike relation. Push-up rule

[F4]

The causal future of the compact source is closed in the globally hyperbolic exterior. In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact

[F5]

Use the causal-curve, chronological-future and temporal-function conventions of the causal-data definition. All relations, closures, interiors and boundaries in this argument are relative to $E$ ; boundary means closure minus interior. Causal exterior, chronological boundaries and Cauchy data

[F6]

On a time-oriented smooth Lorentzian four-manifold, the geodesic with any initial point and tangent exists on an open parameter interval containing zero and is smooth. Use clauses (a) and (b) on $E$ with initial tangent $N_{p}$ , the future timelike orientation vector at $p$ . The geodesic initial-value problem and the exponential map of the metric-selected connection

[F7]

Along a nonconstant geodesic its tangent is nowhere zero, its squared norm is constant, and a causal tangent retains its time orientation. Use clause 3 and Proof 3.2 for the geodesic in [F6]; no geodesic completeness is required. Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed

[A1]

Work in ZFC, including the countable and dependent choice required by the invoked geodesic, compactness and causal interfaces. The Axiom of Choice

Proof

Given: The assumptions in the Statement, with the data and hypotheses specified in [F1].

Goal: Establish the stated conditional claim.

1.1givenA1F1F2F5contradiction

Suppose $p \in C_{t_{R}} \cap I^{+} (A)$ and write $p = Φ (t_{R}, x)$ for some $x \in C_{0}$ . By [F2], $I^{+} (A)$ is open (it is the union of the open chronological futures of points of $A$ ). Continuity of $Φ$ from [F1] then puts $Φ (t_{R} - ε, x)$ in $I^{+} (A) \subset J^{+} (A)$ for all sufficiently small $ε > 0$ . This is a causal contact on $C_{t_{R} - ε}$ , contradicting the minimum property of $t_{R}$ . Hence the whole first cut misses $I^{+} (A)$ . Every causal contact point $p$ on it satisfies $τ (p) > 0$ by the temporal-value conclusion already proved in [F1].

1.2A1F5F6F7construct

For any $p \in E$ , take the geodesic $γ$ of [F6] with $γ (0) = p$ and $γ^{'} (0) = N_{p}$ . By [F7] it is future timelike on a sufficiently small interval about zero. For small $s > 0$ , its restrictions give $γ (- s) ≪ p ≪ γ (s)$ , and smoothness gives $γ (\pm s) \to p$ as $s ↓ 0$ . These segments and their endpoints lie in $E$ , since the geodesic was constructed on that manifold. Thus every neighborhood of $p$ contains points strictly to its timelike past and future.

2.1A1F1F2F3F4F5step 1.1step 1.2algebra∎

If $p \in J^{+} (A)$ , the future points from step 1.2 lie in $I^{+} (A)$ by [F3], so $p \in \overline{I^{+} (A)}$ . Conversely $I^{+} (A) \subset J^{+} (A)$ and [F4] give $\overline{I^{+} (A)} \subset J^{+} (A)$ . Thus $J^{+} (A) = \overline{I^{+} (A)}$ . If $p \in int J^{+} (A)$ , an open neighborhood of $p$ contained in $J^{+} (A)$ contains a point $z ≪ p$ by step 1.2; some $a \in A$ satisfies $a \leq z$ , and [F3] gives $a ≪ p$ . The reverse inclusion $I^{+} (A) \subset int J^{+} (A)$ follows from [F2]. Hence $int J^{+} (A) = I^{+} (A)$ . Using the boundary convention of [F5], $\partial I^{+} (A) = J^{+} (A) ∖ I^{+} (A) = \partial J^{+} (A)$ . The contact points supplied by [F1] belong to that difference by step 1.1, which also proves the whole-cut exclusion and positivity of $τ$ .

Depends on

Used by · 3

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