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Strong Cosmic Censorship at Weak Regularity

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For generic perturbations of Kerr initial data whose maximal globally hyperbolic developments have continuously extendible Cauchy horizons, are those developments nevertheless inextendible by Lorentzian metrics of regularity $C^0\cap W_{\mathrm{loc}}^{1,2}$? More generally, for generic asymptotically flat vacuum initial data, is the maximal globally hyperbolic development inextendible as a weak solution of the Einstein vacuum equations?

Formulation references: [S1], [S2].

Short English statement

For generic perturbations of Kerr initial data whose maximal globally hyperbolic developments have continuously extendible Cauchy horizons, are those developments nevertheless inextendible by Lorentzian metrics of regularity $C^0\cap W_{\mathrm{loc}}^{1,2}$? More generally, for generic asymptotically flat vacuum initial data, is the maximal globally hyperbolic development inextendible as a weak solution of the Einstein vacuum equations?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-34-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-34

The primary theorem excludes C1 future extensions generically in a neighborhood of a rotating Kerr bridge with fixed subextremal parameters. A C0$\cap$W1,2 extension may fail to be C1, so the requested weak-extension exclusion and global generic vacuum assertion do not follow. The theorem gives residual future C0 intersect W1,2-local inextendibility in a sufficiently small weighted-smooth neighborhood of each fixed rotating subextremal Kerr datum. This answers the near-Kerr weak-regularity clause, but not generic arbitrary asymptotically flat vacuum data or zero/extremal rotation. The theorem is local generic C2 inextendibility near a fixed Kerr bridge. It leaves extensions of the weaker C0$\cap$W1,2 class and the problem statement’s global asymptotically flat scope unresolved. This entry records partial support; complete coverage of the question is not established by this catalogue.

Strong Cosmic Censorship at Weak Regularity

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Near each fixed rotating subextremal two-ended Kerr bridge, a dense $G_\delta$ class of smooth vacuum data is claimed to have no future $C^1$ extension; In the same near-Kerr setting, data permitting a continuous nondegenerate future extension with locally square-integrable weak connection are claimed meagre; Near every fixed rotating subextremal Kerr bridge, data with future $C^2$-extendible full developments are claimed meagre, without a symmetry restriction. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.
Fix a complete two-ended Kerr bridge datum $d_*=(h_*,K_*)$ on $\Sigma\cong\mathbb R\times S^2$, of mass $M>0$ and subextremal nonzero rotation $\mathfrak a$. With a smooth end-radius weight $\rho\ge1$, define \[ p_m(u,v)=\sum_{j=0}^m\left(\sup_\Sigma\rho^{1+j}|\nabla_*^ju|_*+ \sup_\Sigma\rho^{2+j}|\nabla_*^jv|_*\right). \] The data space $\mathcal D$ consists of smooth complete positive-definite $h$, with all $p_m(d-d_*)$ finite, satisfying \[ R(h)+(\operatorname{tr}_hK)^2-|K|_h^2=0,\qquad \operatorname{div}_hK-d(\operatorname{tr}_hK)=0. \] Use the relative topology of all these seminorms and $\mathcal U_\varepsilon=\{d:p_{10}(d-d_*)<\varepsilon\}$. An extension has a smooth time-oriented isometric embedding of the full maximal globally hyperbolic development with image open and strictly contained in a connected Lorentzian four-manifold, and a future timelike $C^1$ curve that exits its image at its endpoint. The ambient extension need not satisfy the vacuum equations or global hyperbolicity. For $0<\mathfrak a<M$, the theorem claims some $\varepsilon_0>0$ and a dense $G_\delta$ subset $\mathcal G\subset\mathcal U_{\varepsilon_0}$ whose data have no such future extension with nondegenerate $C^1$ metric. This is Baire genericity in the stated near-Kerr topology.
Result 2.
Fix a complete two-ended Kerr bridge datum $d_*=(h_*,K_*)$ on $\Sigma\cong\mathbb R\times S^2$, of mass $M>0$ and subextremal nonzero rotation $\mathfrak a$. With a smooth end-radius weight $\rho\ge1$, define \[ p_m(u,v)=\sum_{j=0}^m\left(\sup_\Sigma\rho^{1+j}|\nabla_*^ju|_*+ \sup_\Sigma\rho^{2+j}|\nabla_*^jv|_*\right). \] The data space $\mathcal D$ consists of smooth complete positive-definite $h$, with all $p_m(d-d_*)$ finite, satisfying \[ R(h)+(\operatorname{tr}_hK)^2-|K|_h^2=0,\qquad \operatorname{div}_hK-d(\operatorname{tr}_hK)=0. \] Use the relative topology of all these seminorms and $\mathcal U_\varepsilon=\{d:p_{10}(d-d_*)<\varepsilon\}$. An extension has a smooth time-oriented isometric embedding of the full maximal globally hyperbolic development with image open and strictly contained in a connected Lorentzian four-manifold, and a future timelike $C^1$ curve that exits its image at its endpoint. The ambient extension need not satisfy the vacuum equations or global hyperbolicity. For $0<|\mathfrak a|<M$, the source claims an $\varepsilon>0$ for which \[ \{d\in\mathcal U_\varepsilon:\text{a future extension with }\widetilde g\in C^0\cap W^{1,2}_{\mathrm{loc}}\text{ exists}\} \text{ is meagre}. \] The weak metric is continuous and nondegenerate, and its chart components have square-integrable distributional first derivatives. Its weak Christoffel coefficients are locally $L^2$.
Result 3.
Fix a complete two-ended Kerr bridge datum $d_*=(h_*,K_*)$ on $\Sigma\cong\mathbb R\times S^2$, of mass $M>0$ and subextremal nonzero rotation $\mathfrak a$. With a smooth end-radius weight $\rho\ge1$, define \[ p_m(u,v)=\sum_{j=0}^m\left(\sup_\Sigma\rho^{1+j}|\nabla_*^ju|_*+ \sup_\Sigma\rho^{2+j}|\nabla_*^jv|_*\right). \] The data space $\mathcal D$ consists of smooth complete positive-definite $h$, with all $p_m(d-d_*)$ finite, satisfying \[ R(h)+(\operatorname{tr}_hK)^2-|K|_h^2=0,\qquad \operatorname{div}_hK-d(\operatorname{tr}_hK)=0. \] Use the relative topology of all these seminorms and $\mathcal U_\varepsilon=\{d:p_{10}(d-d_*)<\varepsilon\}$. An extension has a smooth time-oriented isometric embedding of the full maximal globally hyperbolic development with image open and strictly contained in a connected Lorentzian four-manifold, and a future timelike $C^1$ curve that exits its image at its endpoint. The ambient extension need not satisfy the vacuum equations or global hyperbolicity. For $0<\mathfrak a<M$, the source asserts that some $\varepsilon>0$ satisfies \[ \{d\in\mathcal U_\varepsilon:\text{a future extension with nondegenerate }C^2\text{ metric exists}\} \text{ is meagre}. \] Equivalently, future $C^2$-inextendible data include a residual subset of this neighborhood. No symmetry condition is imposed on the perturbation.

Outline of the argument

Closed transport tests reduce arbitrary exits to selected interior branches. Exact constrained signed packets are propagated nonlinearly through a double-null region and attached to the full development; the observing geodesic records an obstruction to continuously differentiable extension.

Countably many finite-branch tests cover arbitrary weak exits. Exact-data preparation and a nonlinear double-null comparison produce curvature signals along matched observing states; averaged holonomy detects the weak-connection obstruction and yields the category conclusion.

Closed finite-observation tests are coupled to microlocally separated Kerr packets. Stationary gauges and causal modulation control propagation across compact and far regions, then the two-ended interior estimates produce curvature behavior incompatible with a twice differentiable extension.

Formal verification

Family 264 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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