Global Stability and Asymptotics of Cosmological Spacetimes

Global Stability and Asymptotics of Cosmological Spacetimes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

The de Sitter and Kerr--de Sitter families are backgrounds with positive physical cosmological constant. Nonlinear stability asks whether all constraint-satisfying data sufficiently close in a specified norm have global developments remaining controlled and approaching a described background or asymptotic regime. The sign of the written Einstein equation depends on the curvature convention.

Catalogue wording retained for traceability.

For $\Lambda>0$, do sufficiently small perturbations of de Sitter or Kerr–de Sitter initial data yield global solutions of $\operatorname{Ric}(g)+\Lambda g=0$, and can those solutions be described asymptotically?

Scope and source qualification.

The report already describes established de Sitter and slowly rotating Kerr--de Sitter stability. The broad catalogue wording must not be read as declaring those known cases open; the unresolved parameter and asymptotic extensions need the source hypotheses.

Short English statement

Determine global stability and asymptotics of the stated cosmological backgrounds under sufficiently small admissible perturbations.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved an expanding-background scalar derivative-energy decay estimate and identified its uncontrolled constant mode; no nonlinear cosmological extension.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30006048. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.

An expanding-background energy test.

On a flat periodic spatial torus, take the auxiliary scalar equation \[ \phi_{tt}+3H\phi_t-e^{-2Ht}\Delta\phi=0,\qquad H>0, \] and define $E(t)=\frac12\int(\phi_t^2+e^{-2Ht}|\nabla\phi|^2)dx$. For smooth real solutions, multiplication by $\phi_t$ and periodic integration by parts give \[ E'(t)=-3H\int\phi_t^2dx-H e^{-2Ht}\int|\nabla\phi|^2dx\leq-2HE(t). \] Integrating $(e^{2Ht}E)'\leq0$ yields $E(t)\leq e^{-2Ht}E(0)$.

The nondecaying mode check.

Spatially constant solutions are $\phi(t)=A+B e^{-3Ht}$. Their derivative energy decays, yet $\phi$ approaches an arbitrary constant $A$. An energy that controls derivatives alone therefore cannot force convergence to one fixed background parameter. In a geometric problem the analogous zero or parameter modes must be identified and included in the asymptotic description, not discarded as an energy-estimate detail.

Source limitation.

The scalar estimate does not control the constrained nonlinear Einstein system, gauge, trapping or rotation. The supplied [S2] qualification already records established de Sitter and slowly rotating Kerr--de Sitter stability; those cases are not claimed newly open or newly proved here. For the unresolved extensions the first gap is a suitable tensorial nonlinear energy/modulation estimate in the actual parameter range, with constraints propagated and residual family parameters identified. The curvature-sign convention in the catalogue is not inferred from this scalar model.

Verification and adversarial audit.

Checked the scale-factor derivative, integration-by-parts signs, exponential energy inequality and exact constant spatial mode; preserved the source known stability cases. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked the scale-factor derivative, integration-by-parts signs, exponential energy inequality and exact constant spatial mode; preserved the source known stability cases. First remaining gap: Establish a constraint-compatible tensorial nonlinear energy and parameter-modulation estimate in the source unresolved regime. The inherited source status is preserved; original-target completion remains zero. Additional GPT-6 Astra reviewer (ultra reasoning): Differentiated the scalar energy and integrated the Laplacian by parts on the spatial torus: Eprime=-3H||phi_t||²-H exp(-2Ht)||grad phi||², hence Eprime<=-2HE. Checked the integrating factor and the spatially constant solution A+B exp(-3Ht), whose limiting constant is not controlled by derivative energy. Confirmed the draft preserves the supplied qualification that de Sitter and slowly rotating Kerr-de Sitter stability are already established, and makes no tensorial Einstein or full-range extension claim.

Tools, process and contributions.

Python 3 and local file tools used to read definitions and serialize manuscripts. Calculations are symbolic reasoning; no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Establish a constraint-compatible tensorial nonlinear energy and parameter-modulation estimate in the source unresolved regime.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30006048 (OWR-14298591-012). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mathematical Aspects of General Relativity $2024$, Oberwolfach Reports, DOI 10.4171/owr/2024/36, pp. 2033--2112. Original report and mathematical context. Incorporated report material remains under the source CC BY-SA 4.0 terms unless otherwise noted. https://doi.org/10.4171/owr/2024/36.