--- title: "A stabiliser--Ewens deformation: full Hahn spectrum and a sharp stationary-average chi-square transition" date: 2026-08-09 version: "0.1.0-candidate" doi: 10.5281/zenodo.21864287 pdf: https://github.com/ipitchford/hahn-ewens-mixing-theorem/releases/download/v0.1.0-candidate/hahn-ewens-mixing-theorem-v0.1.0-candidate.pdf repository: https://github.com/ipitchford/hahn-ewens-mixing-theorem archive: https://zenodo.org/records/21864287 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A stabiliser--Ewens deformation: full Hahn spectrum and a sharp stationary-average chi-square transition ## Summary This candidate constructs a family of Markov chains on binary strings and then follows the construction all the way from its stationary law to its full-word spectrum and a sharp asymptotic transition. The count of ones has a beta-binomial stationary distribution. Its complete eigenbasis is the two-parameter Hahn family. That count chain has only $N+1$ states, however, while the labelled binary-word chain has $2^N$ states. The paper proves that the larger chain splits into intrinsic degree layers: degree $\ell$ has eigenvalue $$ \lambda_\ell=\mathbb E[(U-V)^\ell] $$ with multiplicity $\binom N\ell$. Here $U$ and $V$ are beta variables whose parameters come directly from the stabiliser--cycle construction. The main asymptotic theorem asks what those exponentially many modes do to a specific mixing observable. For fixed $\alpha,\beta>0$ and $0 **Candidate status:** anonymous · unrefereed candidate · producer replay passed · > independent reproduction, formal verification, external specialist review, > and editorial peer review not assessed · no parent release. ## From counts to labelled words The construction begins with a random permutation whose cycles receive new binary labels. The two source-colour blocks use different Ewens parameters, and detailed balance constrains the cycle-colouring probabilities. A unique source-blind specialization produces the full two-parameter Hahn family on the count process. Counts alone hide most of the state space. The paper therefore introduces a latent-beta update on each coordinate and filters polynomial functions by their square-free degree. The degree layers are mutually orthogonal and self-adjoint. This yields the complete word-chain characteristic polynomial, including the correct aggregation rule when distinct degrees happen to have the same numerical eigenvalue. The count and word chains share the same absolute spectral radius, but they do not share spectral multiplicities or mixing profiles. That distinction is precisely why the full-word mixing theorem is not a routine corollary of the count spectrum. ## Why the transition occurs on the N/log N scale Write $q_\ell=\mathbb E|U-V|^\ell$. The paper evaluates the positive and negative endpoints separately and proves $$ q_\ell=A_+\ell^{-2c} +A_-\ell^{-(\alpha+\beta-2c)} +o\!\left(\ell^{-2c}+\ell^{-(\alpha+\beta-2c)}\right), $$ with explicit positive constants. If the two exponents coincide, the constants add; there is no logarithmic correction. Even eigenvalues equal these absolute moments, which avoids cancellation in the lower bound. The exact stationary-average identity is $$ \mathcal A_N(t) =\operatorname{tr}(K^{2t})-1 =\sum_{\ell=1}^{N}\binom N\ell\lambda_\ell^{2t}. $$ Above the threshold, a sublinear $N^\gamma$ split controls all small and intermediate degrees, while the remaining binomial mass is defeated by the power-law decay. Below the threshold, one central even degree already diverges. Parity and integer-time rounding contribute only lower-order terms. For the canonical chain the constant becomes $$ \frac{(\alpha+\beta)\log 2}{4\alpha\beta}. $$ At $\alpha=\beta=1$ this is $(\log 2)/2$, recovering the classical constant. ## What is classical and what is candidate-new Diaconis, Lin and Ram already establish the classical trace formula, binomial multiplicities, polynomial eigenvalue decay, the $N/\log N$ mechanism, and stronger classical point-start results. The Schur--Weyl decomposition they use is an operator-independent framework and is not claimed here as new. Relative to the cited public corpus, the plausible increment is the asymmetric stabiliser--Ewens construction, the two-endpoint parameter calculation, and the resulting three-parameter stationary-average theorem. The paper also separates the broad algorithmic construction from the narrower fixed-weight twisted-Burnside formalism. This is a bounded candidate-new claim, not a priority certificate. A directly relevant manuscript attributed to Chenyang Zhong was cited as in preparation and could not be examined. Public release does not resolve that uncertainty. ## What the executable evidence checks The isolated mixing verifier passes under ordinary and optimised Python. Each positive run reports **3,177 explicit predicates**. Each control run reports **107 predicates across ten controls**. The package checks: - three exact routes to the eigenvalue formula; - two algebraically distinct bounded word-kernel constructions; - direct equality between conditional chi-square, matrix trace, and the spectral sum for small exact cases; - the absolute-radius identity and positive even moments; - endpoint regimes with either endpoint dominant and with equal exponents; - finite diagnostics below and above the predicted threshold; and - cause-specific mutations, invalid parameters, and manifested-input corruption. A checksum-bound foundational evidence snapshot preserves the earlier full-word producer replay and its own ordinary, optimised, and negative-control receipts. It is supporting material inside this release, not a parent output. These calculations are producer-side falsification support. They test formula translation, signs, normalization, exact finite kernels, and verifier failure behaviour. They do not prove the endpoint asymptotic, establish independence, or upgrade the assurance of the written theorem. ## What is not established - No unaffiliated rerun or independent implementation has been reported. - No theorem has been formalized in a proof assistant. - No external Markov-chain specialist or journal referee has assessed the paper. - No absolute novelty or priority conclusion is claimed. - No fixed-start, typical-start, worst-case, or total-variation cutoff follows from the stationary-average result. - No conclusion is claimed at the critical transition constant. - Boundary parameter values can lose irreducibility or aperiodicity and are outside the open-region theorem. ## What would most improve the result The highest-value next step is an independent mathematical reconstruction, not a larger producer-side finite grid. A specialist should rebuild the latent beta diagonalisation, endpoint asymptotic, and $N^\gamma$ trace split from the statement alone. Further work could determine the critical-point behaviour, seek typical-start or worst-case analogues, analyse the boundary regimes, and formalize the main proof. ## Package and reuse The public repository contains the 21-page PDF, LaTeX source, exact verifiers, frozen receipts, review records, claim registry, deterministic release tooling, visual abstract, Open Graph card, plain-English transcript, and AI-generated audio. The tagged archive and Zenodo deposit are designed to carry byte-identical release assets with a shared SHA-256 ledger. Original non-code material is dedicated under CC0 1.0 to the extent the repository maintainer holds the relevant rights; original code is MIT. The audio and images explain the work but are not evidence. ## Open directions for follow-up research - Obtain an unaffiliated reconstruction of the latent-beta full-word diagonalisation and the endpoint asymptotic without using the production verifiers. - Determine the transition behaviour at the critical constant log(2)/(2 delta). - Upgrade or refute the stationary-average result for typical starts, specified starts, worst-case chi-square, separation, or total variation in the asymmetric parameter family. - Resolve the exact relationship with the unavailable Ewens-deformation work attributed to Chenyang Zhong and extend the bounded public-corpus search. - Formalize the full-word degree decomposition, endpoint asymptotic, and N-to-the-gamma trace split in a proof assistant. - Analyse parameter-boundary regimes where cycle labels become deterministic and irreducibility or aperiodicity can fail. ## Verification status Anonymous, unrefereed candidate. Internal replay, explicit failure controls, manifests, and package checks pass. The theorem concerns a stationary-weighted average of conditional point-start chi-square distances, not a stationary mixture and not fixed-start, typical-start, worst-case, or total-variation cutoff. No critical-point result is claimed. Same-workflow reviews found no fatal or major mathematical defect after recorded corrections, but no unaffiliated rerun, independent reimplementation, proof-assistant formalization, external specialist review, editorial peer review, or absolute priority determination has occurred. One directly relevant unpublished work could not be assessed. ## Sources and related work - Diaconis, P., Lin, A., & Ram, A. (2025). A curiously slowly mixing Markov chain. arXiv:2511.01245. - Diaconis, P., Lin, A., & Ram, A. (2025). Schur--Weyl duality for diagonalizing a Markov chain on the hypercube. arXiv:2512.23285. - Diaconis, P., & Zhong, C. (2023). Hahn polynomials and the Burnside process. - Diaconis, P. (2005). Analysis of a Bose--Einstein Markov chain. - Ewens, W. J. (1972). The sampling theory of selectively neutral alleles.