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.

The inward null family reaches the future local horizon in short affine time

Statement

Work in ZFC. For the ambient affine geodesics $γ_{ϵ, y}$ with initial vector $n_{ϵ} - ν_{ϵ}$ , there are fixed positive constants C and tau0 such that every sufficiently small positive epsilon has a first crossing $a = 0, b > 0$ at parameter at most $C ϵ < τ_{0}$ . These geodesics remain in a fixed compact tube for $0 \leq λ \leq τ_{0}$ . This is a local assertion in the smooth ambient metric across S, independent of completeness of exterior null geodesics.

Facts & Assumptions

[F1]

The initial outward graph and inward normal have the stated uniform defining-function jets. Small outward normal graphs have controlled inward null initial jets

[F2]

The smooth ambient metric across the bifurcation surface has a smooth geodesic flow on an open domain. The geodesic initial-value problem and the exponential map of the metric-selected connection

[F3]

In the common ambient tube $O$ of the compact surface $S$ , the smooth defining pair $a = v - u$ , $b = v + u$ identifies the future half-sheet as $H_{f}^{+} \cap O = {a = 0, b > 0}$ . A bifurcate future horizon has an oriented defining pair near its section

[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.

Goal: Establish the stated conditional claim.

1.1F1F2F3givenalgebra

The compact initial-data family in [F1] and the open smooth geodesic-flow domain in [F2] give a common tau0 and a smaller compact tube containing all these short geodesics. Smooth dependence and the negative initial a-derivative make $d (a \circ γ) / d λ \leq - 1$ on a common short interval after shrinking. Since $0 < a (0) \leq C_{0} ϵ$ , integration forces a first zero by $C_{0} ϵ$ .

2.1F1F2F3step 1.1algebra∎

On intervals of length at most $C_{0} ϵ$ , smooth dependence gives $∣ d (b \circ γ) / d λ ∣ \leq C_{1} (ϵ + λ)$ : at epsilon zero and lambda zero that derivative vanishes, uniformly over compact S. Hence b changes by $O (ϵ^{2})$ before the crossing, whereas $b (0) \geq c ϵ$ by [F1]. Thus b is positive at that crossing, which is on the future branch by [F3]. Choose C at least C0 and then epsilon small enough that $C ϵ < τ_{0}$ .

Depends on

Used by · 2

Dependency tree · 2 levels

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