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Bibliographic Details
Main Author:	Finks, Christopher
Format:	Recurso digital
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Published:	Zenodo 2026
Online Access:	https://doi.org/10.5281/zenodo.18502790
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Table of Contents:

Five results on sampling permutations σ ∈ S_n with LIS(σ) = k in time polynomial in n and k, produced in under one hour by a multi-stage constraint pipeline designed by Constraint Layer Research. The pipeline uses architecturally distinct constraint engines —  (constraint extraction),   (structural validation),   (synthesis), and AUDIT (adversarial error correction) — with ChatGPT 5.2 as the synthesis substrate and Claude Opus 4.6 as the audit substrate. Results: (1) a poly(n,k) non-uniform sampler with full support on S(n,k) via Robinson-Schensted, uniform partition sampling, and Greene-Nijenhuis-Wilf; (2) a poly(n,k) counting oracle via Gessel's Toeplitz-Bessel determinant; (3) a novel poly(n,k) counting oracle for involutions with bounded LIS, derived by connecting RSK bridge computations to Rains' finite-dimensional SO(2k+1) integral and reducing to a k×k Hankel determinant of Jacobi-exponential moments — this result appears to be original; (4) poly(n,k) bridge computation for fixed-size intermediate RSK shapes via bounded-order differential operators on the Hankel determinant; (5) a structural characterization of why uniform sampling remains open, locating the precise algebraic obstruction at general intermediate RSK shapes. Two mathematical errors were caught during the pipeline's adversarial audit cycle and corrected by the synthesis engine. The synthesis engine halted prematurely four times at literature boundaries, declaring "this requires original mathematics."  The principal investigator directed the audit to override these halts by redefining the termination condition from "unpublished" to "logically blocked," enabling the synthesis engine to derive results it had previously refused to attempt. The key breakthrough — recognizing that Rains' finite-dimensional group integral, not its infinite-dimensional Fredholm scaling limit, yields a direct poly(n,k) algorithmic oracle — emerged through this process. No human mathematical expertise was applied during the pipeline run; only architectural design and real-time constraint enforcement by the principal investigator. The problem was sourced from a Reddit challenge that halted unconstrained frontier models. The constraint architecture — governing when models halt, when they push forward, and how errors are caught and corrected between adversarial roles — is the primary methodological contribution.

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