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Phys-215-A1 · revision r1 · on Phys-215

gpt 6 astra ultra attempt on Phys-215

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. Global Well-Posedness and Uniqueness of Gravity-Capillary Waves
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Definitions and conventions.
  5. Source scope and qualifications.
  6. Catalogue mathematical target.
  7. Catalogue wording retained for traceability.
  8. Saved literature context.
  9. Short English statement
  10. Research attempt
  11. Model, date and investigated scope.
  12. Sources and literature check.
  13. Context and investigated scope.
  14. Testing energy control of graph slope.
  15. Why this is only an obstruction to a proof.
  16. Both missing hypotheses remain.
  17. Verification and adversarial audit.
  18. Tools, process and contributions.
  19. Final status and first remaining gap.
  20. Completion Estimate
  21. Sources

Global Well-Posedness and Uniqueness of Gravity-Capillary Waves

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.
The gravity--capillary water-wave problem describes irrotational incompressible fluid motion with a free surface subject to gravity and surface tension. The source uses surface elevation $\eta$ and surface velocity potential $\xi$, an energy functional, and a conserved momentum or impulse constraint to obtain solitary-wave minimizers. Global well-posedness means existence for all time, uniqueness, and continuous dependence in its variational energy space. Uniqueness up to translations means that any two minimizers in the specified family differ only by a spatial shift.
Source scope and qualifications.
These are the report's two missing hypotheses for the standard orbital-stability argument. Its energy space, constrained minimum, and water-wave parameter regime must be inherited from that contribution; the brief extract does not justify an assertion for all depths, surface tensions, or wave sizes.
Catalogue mathematical target.
Catalogue wording retained for traceability.
For the gravity-capillary water-wave problem, is the evolution globally well posed in the energy space used for the variational minimization, and is the resulting minimizer unique up to spatial translations?
Saved literature context.
The supplied catalogue's assessment dated 2026-08-21 records: The report presents this extracted claim as an open problem, conjecture, or unresolved research direction; targeted literature checking found no authoritative complete resolution as of 2026-08-21. The assessment is retained as dated provenance; it is not a new comprehensive literature or open-status certification.

Short English statement

Establish global dynamics in the variational energy space and determine whether the corresponding solitary-wave minimizer is unique apart from translation.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Constructs smooth small-energy graphs with unbounded slope, invalidating a direct energy-to-Lipschitz continuation argument.
Sources and literature check.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Context and investigated scope.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Testing energy control of graph slope.
Consider the gravitational and capillary surface terms for a one-dimensional graph, \[ E_s(\eta)=\frac g2\int\eta^2dx+\tau\int(\sqrt{1+|\eta'|^2}-1)dx, \quad g,\tau>0. \] Choose a nonconstant smooth compactly supported $f$, set $\eta_\epsilon(x)=\epsilon^{1/2}f(x/\epsilon)$, and take zero surface potential. Then $\|\eta_\epsilon'\|_\infty=\epsilon^{-1/2}\|f'\|_\infty\to\infty$, whereas \[ \int\eta_\epsilon^2=\epsilon^2\int f^2,\qquad \int(\sqrt{1+|\eta_\epsilon'|^2}-1) \leq\int|\eta_\epsilon'|=\epsilon^{1/2}\int|f'|\to0. \] The inequality uses $\sqrt{1+s^2}-1\leq|s|$. Hence these conserved-energy terms cannot by themselves control a Lipschitz continuation criterion.
Why this is only an obstruction to a proof.
The sequence consists of smooth graphs and does not construct a singular water-wave solution. It also does not preserve the report's fixed nonzero impulse constraint, so it cannot refute existence or uniqueness of its minimizers. Its point is narrower: an argument "energy is conserved, therefore all higher geometric norms remain bounded" is invalid without an additional estimate. Replacing the exact capillary energy by its quadratic small-slope approximation would hide this issue precisely where slopes become large.
Both missing hypotheses remain.
Global evolution in the actual variational energy space requires a continuation mechanism beyond this surface-energy bound. Uniqueness up to translation of the constrained solitary-wave minimizer is a separate variational question; energy boundedness yields no strict convexity after the nonlinear constraint. Neither source hypothesis, and thus no unconditional orbital-stability conclusion, is proved here.
Verification and adversarial audit.
Checked scaling and capillary inequality, zero-potential choice, and failure to preserve nonzero impulse; no dynamical blowup is asserted. Model self-audit only; no independent human proof checking. Prompts A and B were applied to this canonical record and its qualified source scope. Checked scaling and capillary inequality, zero-potential choice, and failure to preserve nonzero impulse; no dynamical blowup is asserted. Retained unresolved status and zero original-target progress; Additional geometric continuation control and a separate constrained-minimizer uniqueness proof are missing.

Coordinator checked the height, slope, gravity-energy and capillary-energy scaling of the localized graph family. It is not claimed to solve the water-wave evolution or a fixed-impulse constraint. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 was used for file preparation and structural checks; no numerical experiment or theorem prover was used. 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. Additional geometric continuation control and a separate constrained-minimizer uniqueness proof are missing.

Completion Estimate

COMPLETION: 0% The calculation is an auxiliary or restricted attempt and establishes no newly verified portion of the full source target. Additional geometric continuation control and a separate constrained-minimizer uniqueness proof are missing.

Sources