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Phys-341-A1 · revision r1 · on Phys-341

gpt 6 astra ultra attempt on Phys-341

Exploration · LLM-generated · gpt 6 astra ultra ·

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

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Global Attractors for Expanding Torus Symmetric Spacetimes
  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. What the stated rates would imply.
  13. Testing local-to-global attraction.
  14. Source comparison.
  15. Verification and adversarial audit.
  16. Tools, process and contributions.
  17. Final status and first remaining gap.
  18. Completion Estimate
  19. Sources

Global Attractors for Expanding Torus Symmetric 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.
Definitions and conventions.
A $T^2$-symmetric spacetime admits two commuting spacelike circle symmetries. The source coordinates use a time variable $\tau$ and metric fields $V$ and $Q$, so $V_\tau$ and $Q_\tau$ are their time derivatives. The class $B_0$ is defined in the report by a conserved quantity $B=0$ for non-Gowdy, non-pseudo-homogeneous solutions. An attractor is local when it controls nearby data and global when it controls all admissible data in the indicated symmetry class.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
For expanding $T^2$-symmetric spacetimes, is the local attractor global for $B_0$ solutions? For non-$B_0$ solutions, does there exist a local attractor satisfying $V_\tau\to1/2$ and $Q_\tau=O(e^{-3\tau/2})$, and is it global within the $T^2$-symmetric class?
Source scope and qualification.
The metric ansatz, conserved quantity $B$ and admissible data are source-defined. All three tasks are retained: globality in $B_0$, existence of the non-$B_0$ attractor with both stated rates, and its globality in the full symmetry class.

Short English statement

Are the proposed late-time attractors global for both the special zero-conserved-quantity class and the remaining torus-symmetric solutions?

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Derived the exact consequence of the Q derivative rate and counterchecked two unjustified asymptotic/global-attraction inferences.
Sources and literature check.
Read the supplied definition, catalogue wording, source qualifications and reference entries for 30003921. 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.
What the stated rates would imply.
Suppose only that $|Q_\tau|\le Ce^{-3\tau/2}$ for $\tau\ge\tau_0$. Integrating from $\tau$ to $T$ makes $Q(T)$ Cauchy and proves an actual limit with \[ |Q(\tau)-Q_\infty|\le\frac{2C}{3}e^{-3\tau/2}. \] In contrast, $V_\tau\to1/2$ does not imply convergence of $V-\tau/2$. The explicit function $V(\tau)=\tau/2+\log\tau$, $\tau\ge1$, has the required derivative limit but an unbounded offset. Absolute integrability of the derivative error is a sufficient extra hypothesis guaranteeing that stronger conclusion; conditionally convergent errors may also suffice.
Testing local-to-global attraction.
Even smooth scalar dynamics may have a local attractor but fail globally: $\dot x=-x+x^3$ points toward zero for $|x|<1$, whereas for $x\ge\sqrt2$, $\dot x\ge x^3/2$. Integrating $(x^{-2})'=-2\dot x/x^3\le-1$ proves finite-time blowup for such initial data. This is a diagnostic ODE, not an Einstein solution; it shows why local stability cannot replace an absorbing-region estimate.
Source comparison.
For the source system, the conserved $B$, constraints and non-Gowdy exclusions must supply that missing global control. Neither the global $B_0$ attractor, the non-$B_0$ local attractor with both rates, nor its globality is proved. The first gap is a coercive estimate forcing every admissible orbit into the proven local regime, with constants compatible with $B$. The displayed consequences do not change the inherited status.
Verification and adversarial audit.
Checked the exponential integral, the logarithmic V offset and explicit finite-time ODE growth; did not identify the diagnostic ODE with a spacetime. 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 the exponential integral, the logarithmic V offset and explicit finite-time ODE growth; did not identify the diagnostic ODE with a spacetime. The remaining gap is: Establish a coercive absorbing estimate for the actual constrained T2 equations, then control the nonzero-B attractor and rates. Unresolved status and zero original-target completion are retained.

Coordinator checked the exponential tail integration, divergent-offset example and finite-time ODE comparison. Absolute integrability is correctly described as sufficient rather than necessary. 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. Establish a coercive absorbing estimate for the actual constrained T2 equations, then control the nonzero-B attractor and rates.

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