LemmaStatement: AI-adaptedProof: 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 non-timelike region is confined to the local-rigidity collar

Statement

For the AIK quantitative package there is an open local-rigidity neighborhood $Ω$ of $S_{0}$ carrying a rotational Killing field $Z$ commuting with $T$ . For sufficiently small $ϵ > 0$ depending only on $A$ , the bound $\sup_{S_{0}} ∣ g (T, T) ∣ < ϵ^{2}$ implies ${p \in Σ_{1} : g (T, T) (p) \geq 0} \subset Ω$ . This set includes the stationary-limit points and uses the source Proposition 1.3 non-strict convention. Here rotational means $Z \equiv̸ 0$ and there is a common $t_{0} > 0$ such that its local flow $ϕ^{Z}$ is defined through time $t_{0}$ on $Ω$ and $ϕ_{t_{0}}^{Z} (p) = p$ for every $p \in Ω$ . Individual orbits may have shorter periods or be fixed; no normalization of $t_{0}$ or completeness on all of $M$ is asserted.

Facts & Assumptions

[A1]

Work in ZFC, including the axiom of choice and its inherited countable and dependent choice uses. The Axiom of Choice

[F1]

The stationary package and its fixed quantitative constant are those of Exact AIK global, horizon and quantitative hypotheses.

[F2]

The source estimates imply timelikeness on $1 + ϵ^{1 / 100} \leq r \leq R_{1}$ for sufficiently small $ϵ (A)$ . Quantitative lapse and shift bounds force annular timelikeness

[F3]

The asymptotic end $r > R_{1}$ is timelike. The quantitative asymptotic end is uniformly timelike

[F4]

The explicit External input gives $Ω$ , a nontrivial rotational Killing field $Z$ commuting with $T$ , its common local period $t_{0} > 0$ , and $ρ (A) > 0$ such that $Σ_{1} ∖ Σ_{1 + ρ} \subset Ω$ .

[F5]

Local flows and their time maps use Local and global flows generated by a vector field. The period condition is local to the rigidity neighborhood; no global flow on $M$ is assumed.

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the stated conditional conclusion.

1.1F1F2F3F4given

Choose the horizon parameter below the threshold for [F2] and also so that $ϵ^{1 / 100} < ρ$ . This is possible because $ρ > 0$ is fixed by the source geometric bounds, independently of the subsequently chosen small parameter. By [F2] and [F3], every point with $r \geq 1 + ϵ^{1 / 100}$ has $g (T, T) < 0$ .

2.1step 1.1F1F4F5algebra∎

Therefore a point of $Σ_{1}$ with $g (T, T) \geq 0$ must satisfy $1 < r < 1 + ϵ^{1 / 100} < 1 + ρ$ , so lies in $Σ_{1} ∖ Σ_{1 + ρ} \subset Ω$ . The Killing-field existence, commutation and common-period assertions are exactly [F4], with the local-flow meaning in [F5]; this local application asserts no extension of $Z$ outside $Ω$ .

External input

Import the local rotational-rigidity conclusion of AIK Theorem 1.2 and the quantitative collar inclusion stated in its final proof of Proposition 1.3. The common local period in the rotational conclusion is specified by Proposition 5.2 of the separately cited local-rigidity paper: the time- $t_{0}$ map is the identity on a neighborhood of $S_{0}$ . Both existence and periodicity remain imported premises. Combining them with the separately imported estimates and the local norm/asymptotic calculations proves the stated confinement, without claiming a reconstruction of the local Carleman construction.

Depends on

Used by · 1

Smooth rigidity from a quantitatively small horizon field

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.

External dependencies

S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · §1.1.3 Theorem 1.2, PDF p. 4; final Proof of Proposition 1.3, PDF p. 10

The local rotational Killing field Z commuting with T exists on an open Omega containing S0. With the quantitative bounds fixed, Omega contains Sigma1 minus Sigma_(1+rho) for a rho=rho(A)>0.

Hypotheses: The GR and SBS setting of the source, with its fixed quantitative local geometry; the maximal-section and smallness assumptions are retained in the present application. No global extension of Z is asserted by this local input.

Reading limits: Consulted Theorem 1.2 and the quantitative collar paragraph in the final proof of Proposition 1.3 for the local field and rho(A). These conclusions are imported, not locally proved. The separate local rigidity paper cited as [1] was consulted at the passages recorded in the next reference for the common-period meaning; its Carleman construction was not reconstructed. No global field extension was reviewed.

S. Alexakis, A. D. Ionescu and S. Klainerman, Hawking's local rigidity theorem without analyticity · Abstract, PDF p. 1; Theorem 1.2 and (1.1), PDF p. 2; section 5, Proposition 5.2, PDF p. 17, and its proof, PDF pp. 17–18

The local rotational field Z is nontrivial and commutes with T. Proposition 5.2 specifies its rotational property by one constant t0>0 for which its time-t0 local flow map is the identity on a neighborhood of the bifurcation sphere. This reference supplies the common-period meaning, not a quantitative collar radius.

Hypotheses: A smooth four-dimensional Ricci-flat Lorentzian spacetime with a local regular bifurcate non-expanding horizon and a Killing field T tangent to both branches and not identically zero on the bifurcation sphere. The application retains the full AIK quantitative package.

Reading limits: Consulted the abstract, Theorem 1.2, the paragraph containing (1.1), and Proposition 5.2 with its printed proof for the common-period assertion. The auxiliary lemmas and the local Carleman construction were not reconstructed or independently verified. No quantitative dependence of a collar radius is imported from this reference.

Sources

S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · §1.1.3 Theorem 1.2, PDF p. 4; final Proof of Proposition 1.3, PDF p. 10
S. Alexakis, A. D. Ionescu and S. Klainerman, Hawking's local rigidity theorem without analyticity · Abstract, PDF p. 1; Theorem 1.2 and (1.1), PDF p. 2; section 5, Proposition 5.2, PDF p. 17, and its proof, PDF pp. 17–18