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

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 $0<c<\min(\alpha,\beta)$, define

The stationary-weighted average of the conditional point-start chi-square distances has a sharp transition at

Below that constant the average diverges; above it the average goes to zero. The result is a theorem for the explicitly defined stationary-average trace observable. It is not a worst-case, typical-start, fixed-start, stationary-mixture, or total-variation cutoff theorem, and it says nothing at the exact critical constant.

Candidate status: anonymous · unrefereed candidate · producer replay passed · independent reproduction, formal verification, external specialist review, and editorial peer review not assessed · no parent release.

Summary for specialists

The candidate diagonalises the labelled-word chain by square-free degree, with eigenvalues given by beta-difference moments and binomial layer multiplicities. It then proves a sharp stationary-average chi-square trace transition at the stated constant by combining two-endpoint moment asymptotics with a sublinear degree split. The scope is the explicit observable and open parameter region, not a general cutoff theorem.

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

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

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

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.

Who should care, and why

How to inspect and reproduce the recorded checks

Use the v0.1.0 candidate release or the DOI archive. Check the release manifest before running the ordinary and optimised positive and control paths. Compare the three eigenvalue routes, the two bounded-kernel constructions and the recorded predicate counts; this tests the supplied finite semantics but does not independently reconstruct the asymptotic proof.

What is in the public package

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.

@misc{hahnewensmixingtheorem2026,
  title        = {A stabiliser--Ewens deformation: full Hahn spectrum and a sharp stationary-average chi-square transition},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.21864287},
  url          = {https://doi.org/10.5281/zenodo.21864287},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/hahn-ewens-mixing-theorem/}
}