Theorem Status and Open Censorship Problems

Cosmic censorship is a family of mathematical statements, not one binary proposition. A theorem occupies a cell labeled by matter model, symmetry, asymptotics, initial-data topology, genericity, maximal development, and extension or visibility regularity. A counterexample in one cell need not touch another. The purpose of a status page is therefore to expose the matrix and the missing implications, not to announce that “censorship is true” or “false.”

Evidence cutoff: 2026-08-12. Peer-reviewed theorems are distinguished from conditional results, linear evidence, and current preprints. Later revisions or new exterior-to-interior results can change individual cells without changing the formulation axes.

Required background. Analysis for Mathematical Relativity supplies topology and genericity, while the Einstein equation supplies the MGHD.

Helpful background. Causality and Global Structure supplies weak/strong censorship definitions; PDEs and Green Functions for Gravitation supplies data and weak-solution classes; and Black-Hole Geometry and Mechanics supplies model interiors.

The coordinates of a verdict

Write a censorship statement as

$$(\mathcal E,\mathcal S,\mathcal A, \mathcal D,\mathcal T,\mathcal G, \mathcal C).$$

$\mathcal E$ is the field system, $\mathcal S$ the symmetry restriction, $\mathcal A$ the asymptotics and boundary condition, $\mathcal D$ the data class, $\mathcal T$ its topology, $\mathcal G$ the genericity quantifier, and $\mathcal C$ the conclusion. For weak censorship, $\mathcal C$ may be completeness of a constructed $\mathscr I^+$. For strong censorship it is inextendibility in $C^0$, Lipschitz, Christoffel-$L^2$, $C^2$, or another specified class.

Two results conflict only if all coordinates match and their conclusions are logical negations. Different regularity or topology is not a technical caveat; it defines another conjecture.

Theorem statements by model

Local determinism and maximal development

For sufficiently regular constraint-satisfying Einstein data, local well-posedness and the Choquet-Bruhat–Geroch construction give an MGHD unique up to an isometry fixing the embedded data. This is the starting theorem, not SCC: the MGHD can still admit a proper extension in which the initial hypersurface is no longer Cauchy Choquet-Bruhat–Geroch 1969, theorem and pp. 329–335.

Spherical Einstein–scalar weak censorship

Christodoulou proves that naked outcomes in his spherical massless-scalar $BV\cap L^1$ characteristic class have a positive-codimension instability in a precise two-direction sense Christodoulou 1999, Theorem 4.1, pp. 214–216. This is strong evidence—and a rigorous instability theorem—for that WCC formulation. It is not a theorem for unrestricted smooth vacuum Cauchy data.

Spherical charged-scalar weak censorship

An and Tan’s 2024 v1 preprint claims a charged analogue for the spherical Einstein–Maxwell–charged-complex-scalar system. In the exact characteristic class $BV([0,\infty),dv)\cap L^1((0,\infty),dv/v)$, data outside an exceptional set have complete future null infinity, and every exceptional datum lies in a complex two-dimensional transverse family containing no other exceptional datum An–Tan 2024, Theorem 1.1 and Remark 1, pp. 2–3, and Theorem 9.1, p. 136. The claim is symmetry reduced and unrefereed; its transversality statement is not an unrestricted probability or open-density theorem.

Spherical Einstein–Maxwell–scalar strong censorship

For admissible two-ended asymptotically flat spherical Cauchy data, the MGHD has a nonempty Cauchy horizon and is $C^0$ future extendible, so $C^0$ SCC fails in that model. Luk and Oh prove that a generic class—open in a weighted $C^1$ topology and dense in a weighted $C^\infty$ topology—has the exterior lower tail that forces $C^2$ future inextendibility Luk–Oh 2019 I, Theorems 1.8–1.10 and Luk–Oh 2019 II, Theorems 1.1 and 1.5. Therefore the $C^0$ and $C^2$ formulations have opposite verdicts in the same symmetry-reduced matter class.

Symmetry-reduced vacuum cosmologies

Nonlinear $T^3$ Gowdy theorems establish $C^2$ inextendibility on precise open/dense sets within the two-Killing-field vacuum class Isenberg 2015, §§4–5, pp. 7–11. They demonstrate an SCC mechanism for compact cosmology, but the Gowdy subspace is not open in the full vacuum constraint set.

Rotating vacuum interior continuity

Dafermos and Luk prove that appropriate nonsymmetric spacelike Cauchy data already inside a black hole and approaching subextremal Kerr near timelike infinity develop a nontrivial Cauchy horizon with a continuous metric extension Dafermos–Luk 2025, Theorems 4.2 and 4.24, pp. 50–52 and 64–65 of the open manuscript. This is a peer-reviewed interior $C^0$-stability theorem. It is conditional on the stated interior asymptotics rather than derived from unrestricted exterior Cauchy data.

Countervailing current results

The following results prevent old blanket summaries.

Rough spherical naked solutions. A May 2026 preprint claims nonlinear open- set stability of continuously self-similar spherical Einstein–scalar naked solutions in a localized $C^{1,\alpha}$-adapted Hölder topology Zheng 2026, Definitions 2.7 and 2.11 and Theorem 3.1, arXiv v1. This challenges a WCC formulation in that rough topology while leaving Christodoulou’s BV perturbation statement intact.

Structured characteristic naked solutions. The vacuum exterior construction has a v1 companion interior: the glued spacetime is the maximal development of a complete asymptotically flat characteristic cone and has incomplete future null infinity Shlapentokh-Rothman 2022, Theorem 1.3, pp. 5–6. Its initial data are arbitrarily regular angularly but only $C^{1,s}$ in the null direction Shlapentokh-Rothman 2022, Remark 1.2, pp. 5–6. A July 2026 v1 preprint also constructs nonspherical Einstein–scalar solutions from structured $\kappa$-self-similar incoming data and small outgoing null perturbations, with incomplete $\mathscr I^+$ and a quantified scalar-field obstruction at the singular inner Cauchy horizon An–Wu 2026, Theorem 1.3, pp. 3–4. These are existence results for named characteristic classes, not stable smooth complete-Cauchy-data counterexamples.

Full-subextremal exterior Kerr stability. Hintz’s v2 preprint, revised 2026-08-03, claims nonlinear exterior stability for small smooth vacuum perturbations of every fixed subextremal Kerr background. The simple theorem uses data equal to Kerr outside a compact set; the precise theorem permits a finite partial-polyhomogeneous expansion with a controlled faster-decaying remainder. It supplies the decay needed by the Dafermos–Luk interior theorem for that structured class and states the resulting interior consequence Hintz 2026, Theorem 1.1 and consequence (1), pp. 2–3, and Theorem 13.1, pp. 318–319, arXiv v2. This is a full-subextremal theorem claim, but it remains an unrefereed preprint whose proof uses two same-author companion preprints as black-box inputs. It is not independent confirmation or a result for arbitrary asymptotically flat data.

Vacuum Kerr Lipschitz obstruction. A March 2026 preprint claims curvature blowup and Lipschitz inextendibility near the Kerr Cauchy horizon for interior data satisfying a precise nonlinear Price-law lower estimate Gurriaran 2026, eqs. (4.2.8)–(4.2.9) and Theorem 4.5.1, pp. 51–52 and 55–56, arXiv v2. The lower estimate is expected to be generic but is an assumption in that result. Hintz’s exterior decay theorem does not automatically prove this nonvanishing lower law: a generic higher-regularity SCC result must match the two hypotheses in compatible gauges and data spaces and verify the current preprint dependencies.

de Sitter spectral competition. Neutral scalar modes on fixed Kerr–Newman–de Sitter backgrounds can have $\beta>1/2$ in a near-extremal charged region, permitting finite local scalar energy at the Cauchy horizon, whereas other parameter ranges possess obstructing modes Davey–Dias–Sola Gil 2024, §§2.3.3 and 5. This is linear test-field evidence, not nonlinear Einstein–Maxwell SCC.

These positions are not a two-sided debate with one winner. They occupy different topology, regularity, system, and proof-status cells.

Status matrix

Question	Evidence through cutoff	Missing step
WCC for unrestricted 3+1 vacuum asymptotically flat data	Unresolved open conjecture	Large-data global exterior classification and a precise genericity theorem
WCC for spherical massless scalar in Christodoulou’s BV class	Exceptional naked data are unstable in specified perturbation directions	Translation to other regularities or nonsymmetric data
WCC for spherical Einstein–Maxwell–charged scalar in $BV\cap L^1(dv/v)$ characteristic data	Positive-codimension exceptional-set theorem claimed in a 2024 v1 preprint	Peer review, other topologies, and nonsymmetric data
WCC for rough self-similar spherical scalar data	Open-set naked stability claimed in a 2026 preprint	Independent review, other topologies, nonsymmetric perturbations
Structured characteristic naked solutions	Complete-cone vacuum existence and nonspherical Einstein–scalar existence are claimed in v1 preprints	Open stability, smooth complete Cauchy formation, and genericity in an unrestricted data space
$C^0$ SCC for two-ended spherical Einstein–Maxwell–scalar	False in the theorem’s data class: continuous Cauchy-horizon extensions exist	One-ended collapse and other matter classes are separate
$C^2$ SCC in the same two-ended spherical class	Proved generically by Luk–Oh	Removal of symmetry and transfer to vacuum Kerr
$C^0$ behavior near subextremal Kerr Cauchy horizons	Continuous interior extension is peer-reviewed; Hintz v2 claims the exterior-to-interior consequence for a structured small-data class	Independent verification, broader asymptotic classes, and genericity transfer
Lipschitz SCC near Kerr $i_+$	Claimed conditionally on a nonlinear Price law in 2026	Prove the nonvanishing lower law generically and match it to the exterior theorem’s gauge and data class
KNdS nonlinear SCC	Not settled by existing scalar spectral scans	Coupled nonlinear gravito-electromagnetic interior and genericity theorem
AdS WCC/SCC	Formulation depends on timelike-boundary condition	Global large-data theorem for each named boundary problem

The matrix makes one key point visible: Hintz v2 now supplies a full-subextremal exterior theorem claim and a Dafermos–Luk consequence for its structured data class, but this does not by itself finish SCC. A higher-regularity generic theorem still needs the exact nonvanishing tail asymptotics required by the interior blowup result, compatible gauges and data spaces, and independent verification of the preprint proof chain.

Major open proof interfaces

1. Constraints to exterior. Construct an open or residual class of general large Cauchy data and determine whether its visible exterior is complete.
2. Exterior to event horizon. Prove sharp upper and nonvanishing lower Price laws, including the perturbation sectors relevant to the nonlinear metric.
3. Event horizon to Cauchy horizon. Propagate those bounds through the full rotating interior and classify $C^0$, Lipschitz, weak, and $C^2$ behavior.
4. Genericity transfer. Show that the exceptional set identified by a tail coefficient has the declared topology or codimension in the original constraint manifold.
5. Other asymptotics. Repeat the formulation with de Sitter observer components or an explicit AdS boundary condition rather than importing the asymptotically flat statement.

Van de Moortel’s current survey organizes the modern PDE results and these interfaces Van de Moortel 2025, §§2–7. Its status synthesis is a guide to primary theorems, not a replacement for their hypotheses.

A peer-reviewed theorem resolving one of these questions, a correction or withdrawal of a 2026 preprint, or a counterexample in a specified model and data class could change the comparison. The remaining cases are open questions; they do not imply a universal conclusion about cosmic censorship.

Check the matrix

Consider the sentence: “Continuous Cauchy horizons are stable, so strong cosmic censorship is false.” Give the strongest defensible replacement using current theorems.

Solution

In the two-ended spherically symmetric Einstein–Maxwell–real-scalar data class, MGHDs possess continuously extendible Cauchy horizons, so the $C^0$ inextendibility formulation is false there; nevertheless generic data are $C^2$ future inextendible. For nonsymmetric vacuum data satisfying Kerr-like interior asymptotics, continuous extension across a nontrivial horizon segment is proved. Hintz v2 claims the corresponding exterior-to-interior implication for a structured small-data class, but independent verification, extension to broader data, and the generic higher-regularity verdict remain unsettled. This replacement states system, symmetry, data, regularity, and missing interface.

References

• An, Xinliang, and Hong Kiat Tan. “A Proof of Weak Cosmic Censorship Conjecture for the Spherically Symmetric Einstein–Maxwell–Charged Scalar Field System.” arXiv:2402.16250v1 (2024). arXiv v1.
• An, Xinliang, and Shengrong Wu. “Naked Singularities beyond Spherical Symmetry: Singular Inner Cauchy Horizons for the Einstein–Scalar Field System.” arXiv:2607.07134v1 (2026). arXiv v1.
• Choquet-Bruhat, Yvonne, and Robert Geroch. “Global Aspects of the Cauchy Problem in General Relativity.” Communications in Mathematical Physics 14 (1969): 329–335. DOI.
• Christodoulou, Demetrios. “The Instability of Naked Singularities in the Gravitational Collapse of a Scalar Field.” Annals of Mathematics 149 (1999): 183–217. arXiv.
• Dafermos, Mihalis, and Jonathan Luk. “The Interior of Dynamical Vacuum Black Holes I: The $C^0$-Stability of the Kerr Cauchy Horizon.” Annals of Mathematics 202 (2025): 309–630. DOI.
• Davey, Alex, Óscar J. C. Dias, and David Sola Gil. “Strong Cosmic Censorship in Kerr–Newman–de Sitter.” Journal of High Energy Physics 2024, no. 7 (2024): 113. DOI.
• Gurriaran, Sebastian. “Non-linear Instability of the Kerr Cauchy Horizon near $i_+$.” arXiv:2603.17911v2 (2026). arXiv v2.
• Hintz, Peter. “Nonlinear Stability of Subextremal Kerr Black Holes.” arXiv:2606.28253v2 (2026). arXiv v2.
• Isenberg, James. “On Strong Cosmic Censorship.” Surveys in Differential Geometry 20 (2015): 17–36. arXiv.
• Luk, Jonathan, and Sung-Jin Oh. “Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data I: The Interior of the Black Hole Region.” Annals of Mathematics 190 (2019): 1–111. arXiv.
• Luk, Jonathan, and Sung-Jin Oh. “Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data II: The Exterior of the Black Hole Region.” Annals of PDE 5 (2019): 6. arXiv.
• Shlapentokh-Rothman, Yakov. “Naked Singularities for the Einstein Vacuum Equations: The Interior Solution.” arXiv:2204.09891v1 (2022). arXiv v1.
• Van de Moortel, Maxime. “The Strong Cosmic Censorship Conjecture.” Comptes Rendus. Mécanique 353 (2025): 415–454. DOI. Open PDF.
• Zheng, Weihao. “Nonlinear Stability of Continuously Self-Similar Naked Singularities for the Einstein–Scalar Field Equations I: Main Results.” arXiv:2605.16235v1 (2026). arXiv v1.