Open questions and potential solutions

Write the Gaussian-null horizon data as $(F_0,h_0,\gamma )$. The vacuum constraint is $F_0=\frac{1}{2}\mathrm{\mid }{h}_0{\mathrm{\mid }}_{\gamma }^2-\frac{1}{2}{\mathrm{div}}_{\gamma }{h}_0$. For any global smooth $\psi$, weighted integration gives ${\int }_S{e}^{2\psi }(F_0-\mathrm{\mid }{h}_0+d\psi {\mathrm{\mid }}_{\gamma }^2)d{\mu }_{\gamma }=-\frac{1}{2}{\int }_S{e}^{2\psi }(\mathrm{\mid }{h}_0+d\psi {\mathrm{\mid }}_{\gamma }^2+\mathrm{\mid }d\psi {\mathrm{\mid }}_{\gamma }^2)d{\mu }_{\gamma }\le 0$. The required strictly positive integrand is impossible. A replacement program would need a local continuation theorem compatible with Ricci-flat horizon data, independent control of the horizon generator and transverse data, and a justified global extension. Those are research obligations, not established steps toward a solution.

Prove an applicability obstruction by weighted integration of the independently sourced horizon constraint. Compare it with the exact local theorem hypothesis, then separate this established obstruction from the unresolved replacement program. The calculation is mathematical conditional on the stated geometric equations; it supplies no empirical assertion.

1. Conditionally suppose a smooth degenerate Killing horizon has a nonvanishing generator and nonempty compact connected spacelike section $S$ without boundary. Use stationary Gaussian-null coordinates $g=2dvdr+r^{2}Fd{v}^2+2r{h}_{a}dvd{x}^a+{\mu }_{ab}d{x}^{a}d{x}^b$, with $K={\mathrm{\partial }}_v$. Set $\gamma =\mu {\mathrm{\mid }}_{r=0}$, $F_0=F{\mathrm{\mid }}_{r=0}$ and $h_0=h{\mathrm{\mid }}_{r=0}$. The Ricci-flat constraint is $F_0=\frac{1}{2}\mathrm{\mid }{h}_0{\mathrm{\mid }}_{\gamma }^2-\frac{1}{2}{\mathrm{div}}_{\gamma }{h}_0$. [D-horizon-data, D-horizon-constraint, D-vacuum]
2. For arbitrary $\psi \in C^{\mathrm{\infty }}(S)$ put $w=e^{2\psi }>0$. Integration by parts gives ${\int }_{S}w{\mathrm{div}}_{\gamma }{h}_{0}d{\mu }_{\gamma }=-2{\int }_{S}w⟨d\psi ,h_0{⟩}_{\gamma }d{\mu }_{\gamma }$. Consequently ${\int }_{S}w{F}_{0}d{\mu }_{\gamma }={\int }_{S}w(\frac{1}{2}\mathrm{\mid }{h}_0{\mathrm{\mid }}_{\gamma }^2+⟨d\psi ,h_0{⟩}_{\gamma })d{\mu }_{\gamma }$. [step:1, D-weighted-integration]
3. Subtract ${\int }_{S}w\mathrm{\mid }{h}_0+d\psi {\mathrm{\mid }}_{\gamma }^{2}d{\mu }_{\gamma }$ and complete the square. The result is ${\int }_{S}w(F_0-\mathrm{\mid }{h}_0+d\psi {\mathrm{\mid }}_{\gamma }^2)d{\mu }_{\gamma }=-\frac{1}{2}{\int }_{S}w(\mathrm{\mid }{h}_0+d\psi {\mathrm{\mid }}_{\gamma }^2+\mathrm{\mid }d\psi {\mathrm{\mid }}_{\gamma }^2)d{\mu }_{\gamma }\le 0$. Positive definiteness of $\gamma$ supplies the inequality. [step:2, D-horizon-data]
4. If $F_0>\mathrm{\mid }{h}_0+d\psi {\mathrm{\mid }}_{\gamma }^2$ held everywhere, the continuous positive difference would have a positive minimum on $S$. Since $w>0$ and the Riemannian volume of nonempty $S$ is positive, the left side in step 3 would be strictly positive, a contradiction. Thus no global smooth graph function meets the required inequality. This independently checks the Ricci-flat specialization of the obstruction recorded after Zhou’s equation (2.10). [step:3, D-horizon-data, D-local-theorem]
5. Zhou’s Theorem 3.1 requires this inequality in addition to matching marked transverse jets. Step 4 excludes its application to the proposed Ricci-flat comparison, even if jet matching were independently achieved. This rejects a candidate method; it neither constructs a non-Kerr exterior nor refutes uniqueness. [step:4, D-local-theorem]
6. The remaining program requires new Ricci-flat local rigidity or continuation, followed by a justified global argument. Figueras–Lucietti’s axisymmetric theorem cannot provide the missing symmetry by its own conclusion. Its explicit symmetry assumption and qualifications must be matched before any application. No such matching or global extension is established here, so the solution extent remains none and its status unresolved. [step:5, D-axisymmetric-comparison, D-question-formulation]

• D-vacuum: The assigned question concerns smooth four-dimensional stationary Lorentzian geometry satisfying $\mathrm{Ric}(g)=0$. In the comparison convention $\mathrm{Ric}(g)=\mathrm{\Lambda }g$, this is $\mathrm{\Lambda }=0$. This is a stipulated mathematical hypothesis, not an observational premise. Source: self-contained; Assigned question; proof step 1. Reading limits: No physical measurement or derivation of the vacuum model is claimed.
• D-horizon-data: For the conditional calculation, $S$ is a nonempty compact connected smooth spacelike section without boundary of an actual smooth degenerate Killing horizon with a nonvanishing normalized generator. The stationary Gaussian-null metric has the form displayed in step 1; its horizon section metric $\gamma$ is positive definite. Source: https://arxiv.org/html/2611.11495v1; §2 opening paragraph and equation (2.1); Theorem 3.1. Reading limits: Read these definitions and the theorem statement in HTML. The Gaussian-null coordinate construction was not independently reconstructed. Existence of this horizon structure for every spacetime in the assigned question remains unproved.
• D-horizon-constraint: In these conventions, Einstein horizon data satisfy $F_0=\mathrm{\Lambda }+\frac{1}{2}\mathrm{\mid }{h}_0{\mathrm{\mid }}_{\gamma }^2-\frac{1}{2}{\mathrm{div}}_{\gamma }{h}_0$, where divergence of the one-form means divergence of its metric dual. Source: https://arxiv.org/html/2611.11495v1; §2, paragraph immediately preceding equation (2.10), identifying the intrinsic vr equation. Reading limits: Read the displayed constraint, weighted identity and following sign conclusion. The constraint is an explicit imported input; a full coordinate derivation of the Einstein equations is not claimed.
• D-weighted-integration: On compact boundaryless Riemannian $S$, ${\int }_{S}w\mathrm{div}Xd\mu =-{\int }_S⟨dw,X⟩d\mu$. A local justification uses $\mathrm{div}X=\mathrm{\mid }\gamma {\mathrm{\mid }}^{-1\mathrm{/}2}{\mathrm{\partial }}_a(\mathrm{\mid }\gamma {\mathrm{\mid }}^{1\mathrm{/}2}{X}^a)$: integrate compactly supported chart terms and sum a finite smooth partition of unity. Terms differentiating the partition cancel because its sum is one. Apply the product rule to $wX$ and use $d(e^{2\psi })=2e^{2\psi }d\psi$. Source: self-contained; Local integration argument supplied here; proof step 2. Reading limits: Elementary smooth-manifold integration with Riemannian densities; no orientation or horizon field equation is used in this input.
• D-local-theorem: Zhou’s Theorem 3.1 assumes smooth Einstein metrics with common Einstein constant, identified compact connected boundaryless spacelike sections, nonvanishing normalized degenerate Killing generators, matching full marked transverse jets, and one global smooth $\psi$ satisfying $F_0>\mathrm{\mid }{h}_0+d\psi {\mathrm{\mid }}_{\gamma }^2$. Its conclusion is a marked local isometry on a two-sided whole-section neighborhood. Source: https://arxiv.org/html/2611.11495v1; Theorem 3.1 and equation (3.1); compare §2 equation (2.10). Reading limits: Read the exact theorem statement and graph-condition passages. The continuation proof was not audited or used to prove an isometry here; only its required hypothesis is tested.
• D-question-formulation: For formulation, I⁺-regularity includes completeness of the stationary Killing field, global hyperbolicity of the domain of outer communications, and a connected spacelike acausal hypersurface containing the selected end, with closure a compact part plus finitely many asymptotic ends and boundary meeting each relevant future-horizon generator exactly once. The full definition is retained by reference. Source: https://arxiv.org/pdf/0806.0016; Definition 1.1 and equation (1.1), PDF/printed p.2. Reading limits: Freshly read this definition in web-extracted PDF text. Other locators in the verbatim historical source record below were not thereby read. These global conditions formulate the question and are not used in the local obstruction.
• D-axisymmetric-comparison: Figueras–Lucietti’s Theorem 1 concerns four-dimensional asymptotically flat stationary axisymmetric rotating vacuum black holes with connected degenerate horizon and non-toroidal sections. Footnote 5 explicitly notes further technical assumptions, principally analyticity. It is contextual comparison, not a supplier of missing symmetry. Source: https://arxiv.org/pdf/0906.5565v4; Introduction, PDF p.3, Theorem 1 and footnote 5. Reading limits: Freshly read the theorem and footnote in extracted PDF text, plus discussion excerpts on PDF pp.18–19. Screenshot retrieval failed. No global applicability or harmonic-map proof audit is claimed.

Decision: accept

All six steps are valid within the stated conditional horizon setup. The integration-by-parts sign and completed-square identity are correct. Strict positivity everywhere contradicts the resulting nonpositive integral. This excludes the proposed application of Zhou’s theorem, even with matched transverse jets. The direction correctly retains solution extent none and status unresolved; acceptance supplies no uniqueness proof or acceptance of the cited continuation proofs.

Dependency review: D-vacuum is a stipulated geometric equation, not an empirical premise. D-horizon-data, D-horizon-constraint and D-local-theorem match the inspected Gaussian-null setup, constraint and theorem statement in Zhou, §2 and Theorem 3.1. Only the constraint and required graph inequality support the obstruction; no continuation conclusion is invoked. D-weighted-integration supplies a valid non-circular local proof using chart integration and a partition of unity, including nonorientable sections through densities. Steps 2–4 follow for every global smooth ψ. D-question-formulation faithfully summarizes Chruściel–Costa, Definition 1.1, with the full definition retained by reference. D-axisymmetric-comparison correctly records the explicit symmetry assumption and analytic qualification in Figueras–Lucietti, Theorem 1 and footnote 5; it supplies no missing symmetry. Formulation and argument dependencies are covered, with no logical cycle or use of an open conclusion as a premise. The calculation is mathematical conditional on Ricci-flat geometry. The original article supplies question attribution only.

Review reading limits: Freshly read the complete assigned direction, its six proof steps and seven immediate dependency records. Read the frozen article’s extracted pp.1–6 and inspected its p.5 image to check §1.3 attribution; this was not a full original-article assessment. Independently read Zhou’s abstract, §2 setup, constraint and equation (2.10), and the full Theorem 3.1 statement in versioned HTML. Read Figueras–Lucietti’s Theorem 1 and footnote 5, and Chruściel–Costa’s Definition 1.1 and equation (1.1), in web-extracted PDF text. Independently checked the weighted integration and square completion. Did not audit the continuation, harmonic-map or globalization proofs. Historical reading claims preserved in the draft’s source records were not inherited.