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.

Time reversal supplies a source-normal generator without interior source focal points

Statement

Work in ZFC. Let $(E, g, N)$ be a smooth globally hyperbolic time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ , and let $τ : E \to R$ be temporal, with past-directed timelike metric gradient. Let $A \subset τ^{- 1} (0)$ be a nonempty compact smooth embedded two-manifold without boundary with positive-definite induced metric. Use the data and hypotheses of the earliest-cylinder-contact construction on this same exterior, source and temporal function, and let $p_{R}$ be one of its earliest contact points. All chronological relations and boundaries below are relative to $E$ .

There is a nonconstant future affine null geodesic from $A$ to $p_{R}$ , wholly in $\partial_{E} I_{E}^{+} (A)$ , initially normal to $A$ and without a focal point to $A$ strictly before $p_{R}$ , with focality measured from $A$ toward $p_{R}$ . The statement supplies both the direction and the focal orientation correctly; it does not yet choose the outgoing source ray.

Facts & Assumptions

[F1]

Use the full data and hypotheses of the first-contact chronological-boundary lemma on this same $(E, g, N)$ , source $A$ and temporal function $τ$ . Its earliest contact satisfies $p_{R} \in \partial_{E} I_{E}^{+} (A)$ and $τ (p_{R}) > 0$ . First contact lies on the chronological and causal future boundary

[F2]

The compact-target generator theorem supplies a null segment on a chronological boundary, normal at the compact surface and without focal points from that surface. Compact-target null generators and the required endpoint conditions

[F3]

Use the causal-curve, chronological-relation, boundary, temporal-function and global-hyperbolicity definitions on $E$ . Reversing the time orientation replaces $N$ by $- N$ while retaining the metric and topology. Causal exterior, chronological boundaries and Cauchy data

[F4]

A spacelike two-surface is smooth, embedded, nonempty and without boundary, with positive-definite induced metric. Focality from such a surface is rank loss of its normal screen Jacobi map along the geodesic parametrized away from that surface. Null expansions, screen quotients and focal data

[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 data and hypotheses in the Statement.

Goal: Obtain the source-normal future null boundary segment with the stated source-oriented focal exclusion.

1.1givenA1F3

Replace $N$ by $- N$ and write $I_{r e v}^{-}$ for the chronological past in that orientation. Reversing a timelike curve parameter gives $I_{r e v}^{-} (A) = I_{E}^{+} (A)$ , and the boundaries agree because the topology is unchanged. Strong causality is unchanged by reversing curve parameters, and each reversed causal diamond is an original diamond with its endpoints exchanged, so global hyperbolicity is preserved. The function $- τ$ is temporal for $- N$ : its gradient has the same negative squared norm as that of $τ$ , and $d (- τ) (- N) = d τ (N) > 0$ .

2.1givenA1F1F2F4step 1.1

By [F1] and step 1.1, $p_{R} \in \partial_{E} I_{r e v}^{-} (A)$ and $- τ (p_{R}) < 0 = \min_{A} (- τ)$ . The same $A$ is a compact spacelike two-surface in the reversed orientation, since its smoothness, dimension, absence of boundary and induced metric have not changed. Thus every hypothesis of [F2] holds with target $A$ and temporal function $- τ$ . It supplies a null affine segment from $p_{R}$ to a point $a \in A$ , future-directed in the reversed orientation, contained in $\partial_{E} I_{r e v}^{-} (A)$ , orthogonal to $A$ at $a$ , and without an interior focal point measured from $A$ toward $p_{R}$ . It is nonconstant because $τ (p_{R}) > 0 = τ (a)$ .

3.1F2F3F4step 1.1step 2.1algebra∎

Reverse the affine segment parameter. Its tangent changes sign and its covariant acceleration remains zero, so it is now a future affine null segment from $a \in A$ to $p_{R}$ in the original orientation. Its image lies in $\partial_{E} I_{E}^{+} (A)$ by step 1.1, and sign reversal preserves orthogonality at $a$ . The focal exclusion in [F2] was already measured from $A$ toward $p_{R}$ , which is exactly the source direction of the reparametrized segment under [F4]; no exchange of source and target focal data is made. A further positive affine scaling only reindexes this normal screen Jacobi map and preserves its rank, as well as the segment image and boundary containment. Selection of the outgoing source ray remains a separate input.

Depends on

Used by · 3

Dependency tree · 2 levels

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