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.

Exact assembly of the global I-plus regularity assumptions

Statement

The smooth global comparison data imply the following complete $I^{+}$ -regularity package relative to the selected end: the stationary field is complete on $M$ ; $D = I_{M}^{+} (M_{e x t}) \cap I_{M}^{-} (M_{e x t})$ is globally hyperbolic; and there is a spacelike connected hypersurface $S \subset D$ , acausal in $M$ and containing the end, whose ambient closure is a topological manifold with boundary, consisting of a compact core and finitely many asymptotic ends. Its boundary lies in $E^{+} = \partial_{M} D \cap I_{M}^{+} (M_{e x t})$ and meets every generator of $E^{+}$ exactly once. These are all the clauses of the declared $I^{+}$ -regularity predicate. No orbit-space coordinate system, analyticity or local Killing-field extension is part of this definition.

Facts & Assumptions

[F1]

The supplied global comparison data impose the full end, complete-flow and I-plus-regularity predicates, together with a global axial action and connected event horizon. Smooth global data for the axisymmetric comparison

[F2]

The comparison definition specifies the compact-core-plus-finite-ends closure, ambient boundary, once-only generator cross-section and end decay explicitly. Staticity, axisymmetry and black-hole regularity hypotheses

[F3]

Chronological reachability and global hyperbolicity refer to the stated domains, and the distinction between ambient and restricted causal curves is retained. Causal exterior, chronological boundaries and Cauchy data

Proof

Given: The data and assumptions in the Statement.

Goal: Establish the claimed conditional conclusion.

1.1F1F2F3given

Expand the $I^{+}$ -regularity requirement in [F1] using [F2]. It gives completeness of the stationary field on all of $M$ , global hyperbolicity of the specific $D$ , and the spacelike connected ambient-acausal $S$ containing the specified end. In particular, compactness applies to the core of the completion and does not remove the stated asymptotic ends.

2.1F2F3step 1.1∎

The remaining clauses in [F2] identify the boundary in the ambient topology, retain the factor $I_{M}^{+} (M_{e x t})$ in $E^{+}$ , and require exactly one intersection with every generator. Together these are precisely the displayed full predicate, with no hypothesis about a quotient coordinate chart. A global uniqueness theorem using this predicate must supply its own global reduction; expanding the definition neither proves nor assumes that reduction.

Depends on

Used by · 1

Axisymmetric vacuum uniqueness with complete global hypotheses

Dependency tree · 2 levels

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