gpt 6 astra ultra attempt on Phys-114

Perturbation theory for exactly solvable combinatorial problems

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 notation.

Exactly solvable probabilistic models admit algebraic formulae for their distributions. In last-passage percolation, geometric or exponential site-weight laws lead to tractable formulae. A perturbation changes the weight distribution; a universality result would show persistence of a normalized fluctuation limit for a declared class of such changes.

Catalogue wording retained for traceability.

The asymptotic behavior of a variety of combinatorial problems has now been analyzed in great detail. Here we have in mind Ulam's problem for the length of the longest increasing subsequence, the tiling problem for the Aztec diamond, the hexagon tiling problem, and the last passage percolation problem, amongst many others. In all cases, in an appropriate scaling limit, the statistical fluctuations in the systems at hand are described by random matrix theory. The asymptotic analysis, however, depends in a critical, and rigid way, on the underlying probability measures for the systems. For example, the analysis of the last passage percolation problem requires the waiting time $w_i$ at each site $i$ to be either geometrically or exponentially distributed (Johansson): if the statistics of the waiting times is neither geometric nor exponential, the analysis fails completely. What happens if the geometric distribution, say, is slightly perturbed? The challenge here is to develop an effective perturbation theory for such systems. One expects that the random matrix behavior of the fluctuations should persist. Related work on this problem has been done by Baik-Suidan and Bouchard-Martin (cf. Problem 10).

Scope and source qualification.

The inherited catalogue status is solved; this historical statement is retained without independently verifying that settlement.

Short English statement

Build an effective perturbation theory that preserves random matrix fluctuation behavior beyond exactly solvable probability laws.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved a conditional distribution-transfer lemma for perturbations vanishing faster than the fluctuation-scale sensitivity bound. The catalogue remains source-solved; its full settlement is not reconstructed or verified here.

Sources and literature check.

Read the supplied complete definition, catalogue wording, scope qualifications and source list for this canonical ID. The original-source citations are inherited as supplied references; no fresh primary-source or novelty search was performed for this attempt. No current open-status claim is inferred from this calculation.

A coupling estimate, retaining the source-solved label.

The catalogue marks this historical programme solved; the present calculation neither audits that settlement nor reopens it. For $n\times n$ directed last-passage arrays $w,\widetilde w$, let $D_n=\max_{ij}|w_{ij}-\widetilde w_{ij}|$. Every admissible path has $2n-1$ sites, hence \[|L_n(w)-L_n(\widetilde w)|\le(2n-1)D_n. \] For a deterministic scale $a_n>0$, if $nD_n/a_n\to0$ in probability and $(L_n(w)-b_n)/a_n$ has a limiting distribution $F$, then the normalized perturbed passage time has the same limit at every continuity point of $F$. Indeed, for each $\eta>0$, its distribution function is trapped between the unperturbed ones at $x-\eta$ and $x+\eta$, up to the probability that $(2n-1)D_n/a_n>\eta$. Let $n\to\infty$, then $\eta\downarrow0$.

An explicit vanishing-perturbation regime.

If each perturbed weight is $\widetilde w_{ij}=w_{ij}+\delta_n\xi_{ij}$, with $|\xi_{ij}|\le1$ almost surely, then $D_n\le\delta_n$. At scale $a_n=n^{1/3}$, the transfer works for $\delta_n=o(n^{-2/3})$. Independence of the perturbations is unnecessary for this deterministic comparison. A constant shift can instead be absorbed exactly into $b_n$, since it adds $(2n-1)\delta$ to every path.

First gap.

For a fixed nontrivial change of the weight law, the crude comparison is order $n$, much larger than the fluctuation scale. Proving cancellation or a new centering does not follow from a sup-norm bound. Thus this elementary conditional transfer lemma explains a limited stability window but does not reconstruct the full effective perturbation theory or establish universality for fixed perturbed geometric/exponential laws. The inherited solved status is preserved separately from this unresolved attempt's limited scope.

Verification and adversarial audit.

Checked path count, deterministic coupling inequality, distribution-function sandwich, required n^{-2/3} window and explicit preservation of source-solved status. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked path count, deterministic coupling inequality, distribution-function sandwich, required n^{-2/3} window and explicit preservation of source-solved status. Scope remains limited: Control fixed-distribution perturbations after model-dependent recentering, beyond the crude pathwise sup-norm estimate.

Coordinator checked the distribution-function squeeze and vanishing-perturbation scale. The original solved catalogue label is explicitly retained independently of the new unresolved attempt. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local shell used to read supplied files and serialize this manuscript; mathematics derived symbolically in the text, with no numerical experiment or formal proof assistant. 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. Control fixed-distribution perturbations after model-dependent recentering, beyond the crude pathwise sup-norm estimate.

Completion Estimate

COMPLETION: 0% The calculation is an elementary benchmark or conditional reduction; it establishes no claimed new portion of the complete source target.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 8000013 (AMR-079-0013). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Percy Deift, Some open problems in random matrix theory and the theory of integrable systems $2007$, arXiv:0712.0849, Problem 13. https://arxiv.org/abs/0712.0849.