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Phys-336-A1 · revision r1 · on Phys-336

gpt 6 astra ultra attempt on Phys-336

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Nonlinear Instability of Anti–de Sitter Spacetime
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Definitions and conventions.
  5. Target retained from the canonical source catalogue.
  6. Catalogue wording retained for traceability.
  7. Source scope and qualification.
  8. Short English statement
  9. Research attempt
  10. Model, date and investigated scope.
  11. Sources and literature check.
  12. Testing small stationary black holes as a shortcut.
  13. The central countercheck.
  14. Remaining scope.
  15. Verification and adversarial audit.
  16. Tools, process and contributions.
  17. Final status and first remaining gap.
  18. Completion Estimate
  19. Sources

Nonlinear Instability of Anti–de Sitter Spacetime

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.
Definitions and conventions.
Anti-de Sitter space solves the vacuum Einstein equations with negative cosmological constant, $\operatorname{Ric}(g)=\frac{2\Lambda}{D-2}g$ in spacetime dimension $D$, with $\Lambda<0$. Its conformal boundary is timelike; reflecting boundary conditions prevent energy from escaping there. Arbitrarily small perturbations means that for every positive size threshold in a specified initial-data norm one seeks smaller constraint-compatible data. A black-hole region is defined relative to causal communication with the conformal boundary.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
Can arbitrarily small perturbations of anti–de Sitter initial data, evolved by the vacuum Einstein equations with reflecting conformal-boundary conditions, form black-hole regions at sufficiently late times?
Source scope and qualification.
This target is the vacuum problem. Scalar-matter examples or resonant approximations, including IDs 30002150 and 30002920, are related mechanisms rather than proofs of the stated vacuum instability. The norm and boundary regularity must be specified in a theorem.

Short English statement

Can sufficiently late black holes arise from vacuum anti-de Sitter perturbations of arbitrarily small size under reflecting boundary conditions?

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Showed small-mass stationary AdS black holes converge only away from the center and do not furnish the requested small regular collapse data.
Sources and literature check.
Read the supplied definition, catalogue wording, source qualifications and reference entries for 30003912. The historical source/context and inherited status are retained; no fresh literature or novelty certification is claimed. No external theorem is used without proof in this attempt; explicit model assumptions are distinguished from facts about the source model.
Testing small stationary black holes as a shortcut.
In four dimensions write the Schwarzschild--AdS lapse as $f_M(r)=1+r^2/L^2-2M/r$, with $L>0$, $M>0$. Its positive horizon radius $r_h$ solves \[ 2M=r_h(1+r_h^2/L^2). \] The right side is strictly increasing for $r_h\ge0$, so there is a unique positive solution and $r_h\le2M$. Dividing by $2M$ and taking $M\downarrow0$ gives $r_h/(2M)\to1$, because $r_h^2/L^2\to0$. At every fixed positive radius the lapse converges to the AdS lapse.
The central countercheck.
This convergence is not uniform down to the center: at $r=2M$, the lapse difference $2M/r$ equals one for every $M$. The geometry also already has a horizon and cannot serve as a smooth global perturbation of a horizon-free AdS slice by merely citing small mass. Thus the existence of arbitrarily small stationary black-hole parameters does not prove formation from arbitrarily small regular initial data in the requested norm.
Remaining scope.
A proof needs constraint-compatible smooth vacuum data, reflecting boundary control and nonlinear evolution long enough to create a black-hole region. The calculation uses neither scalar matter nor a resonant approximation, and proves no instability in any initial-data topology. The first gap is the actual dynamical concentration mechanism with controlled small vacuum initial norm. The known lapse family here is only a concrete test of an invalid shortcut, not new progress on the source vacuum problem.
Verification and adversarial audit.
Checked uniqueness and scaling of the horizon root, fixed-radius convergence and its failure at radii proportional to mass. This is a GPT-6 Astra Ultra self-audit, not independent human verification. Prompt A drafting followed by prompt B scope/dependency/boundary self-audit. Checked uniqueness and scaling of the horizon root, fixed-radius convergence and its failure at radii proportional to mass. The remaining gap is: Construct smooth small vacuum data and control reflecting evolution until a new trapped/black-hole region forms. Unresolved status and zero original-target completion are retained.

Coordinator checked monotonicity of the horizon equation, the small-mass ratio and the nonuniform central lapse difference. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 and local file-reading/writing tools for manuscript serialization and structural checks; mathematical calculations were symbolic reasoning, with 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. Construct smooth small vacuum data and control reflecting evolution until a new trapped/black-hole region forms.

Completion Estimate

COMPLETION: 0% The proved calculation is an elementary auxiliary result or a test of a restricted strategy; no new portion of the full source target is established.

Sources