---
title: "Untouched subsets and a counterexample to the biased-transposition limit-profile conjecture"
date: 2026-09-11
version: "1.0.1-candidate"
doi: 10.5281/zenodo.22708622
pdf: https://github.com/ipitchford/biased-transposition-profile-counterexample/releases/download/v1.0.1-candidate/biased-transposition-profile-counterexample-1.0.1-candidate.pdf
repository: https://github.com/ipitchford/biased-transposition-profile-counterexample
archive: https://zenodo.org/records/22708622
license: CC0-1.0
status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)
---

# Untouched subsets and a counterexample to the biased-transposition limit-profile conjecture

## Summary

An overall count can miss a slow subgroup. In a shuffle where half the cards are selected less often, this candidate uses one event—at least two slow labels are still fixed—to rule out a proposed formula for the transition towards randomness.

At one specified point, that event forces a distance of at least $0.710647716\ldots$ from a random permutation. The conjectured formula predicts $0.681595297\ldots$. The gap is strict. The result concerns the formula in Nestoridi–Yan's arXiv version 1; the established cutoff theorem remains intact.

## Summary for specialists

Let $N$ be even. Half the labels have selection probability $b/N$ and half $(2-b)/N$, for fixed $0<b<1$. Each discrete step samples two labels independently, retains identity self-samples, and transposes distinct labels. At

$$t_N=\left\lfloor\frac{N(\log N-\log6)}{2b}\right\rfloor,$$

the candidate proves

$$\liminf_{N\to\infty,\,2\mid N}d_N(t_N)\ge B_0=\frac32e^{-1/2}-4e^{-3}.$$

The proposed value is $D_0=5/(2e)-13e^{-4}$, and exact arithmetic certifies

$$\frac{29}{1000}<B_0-D_0<\frac{291}{10000}.$$

Thus Conjecture 1.6 in arXiv:2409.16387v1 fails at $s=\log6$ for every fixed slow weight in the stated range.

## Technical account

An untouched slow label remains fixed. At the chosen time, the number of untouched slow labels converges to a Poisson law of mean three. In stationarity, the number of fixed slow labels converges to a Poisson law of mean one half. The difference between the probabilities of having at least two such labels gives the displayed full total-variation lower bound. Returned labels need no joint limit theorem for this argument.

A second proof uses only eleven untouched factorial moments and ten stationary factorial moments. The rational witness $5730077809/8174960640$ exceeds the conjectured value by more than $19/1000$. A general deterministic-subset theorem then produces one-sided Poisson tail bounds for sparse and continuously distributed slow weights.

## Evidence, assurance and limitations

The written argument is the main evidence for the asymptotic claims. Exact arithmetic verifies the strict inequalities and four finite event bounds. An implementation-diverse replay within the package checks eight families on all $8!$ states and all 61 stored times, plus smaller integer and spectral checks. The final suite rejects 23 deliberately corrupted inputs or publication fields.

Five internal editorial roles and one bounded confirmation review inspected frozen packages. The confirmation found no new scientific blocker and required deterministic repairs to publication records and formula correspondence. Those reports are available with their original decisions and subsequent disposition. External review, independent reproduction, formal proof and historical priority remain unestablished.

The general $t_{\rm mix}=t_*+O(n)$ upper bound remains open here. Larger inherited numerical experiments are preserved for provenance and were not fully replayed. The manuscript's article and supplementary dossier are explicitly separated.

## Relationship to earlier work

Nestoridi–Yan's arXiv v1 states the targeted profile conjecture; the official 2025 FPSAC poster restates it. Their proceedings article treats cutoff and spectral results and does not discuss the profile. Teyssier's uniform-transposition profile supplies the classical benchmark. The source audit also records a forthcoming journal listing for Nestoridi–Yan; no public final text was located in the bounded search.

The earlier Evidence Press release on bounded product weights concerns cutoff and a window bound. This candidate addresses a specific profile and uses an observable subset event. Its proof does not depend on the earlier Evidence Press candidate.

## Who should care, and why

| Audience | Potential use | Qualification |
|---|---|---|
| Mixing-time researchers | Inspect a precise profile counterexample and subset lower-bound method. | The replacement profile is unresolved. |
| Sampling researchers | Investigate diagnostics sensitive to slow subgroups. | No guarantee transfers automatically to other chains. |
| Research agents | Reuse exact inequalities, model conventions and replay controls. | Preserve the distinction between finite checks and asymptotic proof. |

## Why the problem matters

A cutoff says that a transition is abrupt; a profile describes its detailed shape. A formula for that shape must survive every observable event. This example shows how a small amount of class information can challenge a prediction suggested by an aggregate statistic.

## How to inspect or reproduce the recorded checks

Extract the versioned archive and run `python3 code/verify_manifest.py` before changing any files. Run `python3 code/verify_exact.py`, then install `requirements-numerical.txt` and run `python3 code/verify_numeric.py --input source_inputs/biased_transposition`. The hostile suite is `python3 code/test_negative_controls.py`; repeat with `python3 -O` to check that optimization does not remove acceptance gates.

`python3 code/verify_claim_surface.py --documents` additionally checks headline formulas and regenerates the TeX and PDF text. This route requires the recorded Pandoc, Tectonic and Poppler tools. Read Section 2 for the event proof and Section 3 for the general subset argument.

## The most valuable next projects

1. Inspect the full proof and pinned source correspondence independently.
2. Determine the correct limiting profile for the two-class walk.
3. Find matching upper bounds for general bounded product weights.

## Who might contribute

Researchers in Markov-chain mixing, random permutations and asymptotic probability can examine the proof and its novelty. Independent verifier runs would add a separate evidence record.

## What is in the evidence package

The archive contains the 13-page candidate paper, six-page audit, Markdown and TeX, code, exact contract, replay reports, internal reviews, complete manifest and component licences. Original prose and research data are CC0-1.0; original code is MIT. Inherited inputs retain their recorded rights and are not relicensed.




## Open directions for follow-up research

- Determine a replacement limit profile.
- Establish a matching upper bound at the untouched-label centre for general bounded product weights.
- Obtain external scrutiny of the written proof and contribution-specific novelty.

## Research process, metrics and reusable methods

This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work.

- Work ID: ep-work:biased-transposition-profile-counterexample
- Attempt and metric receipts: ep-attempt:biased-transposition-profile-counterexample-assurance-publication: published / positive; scope assurance-through-publication; target Close deterministic confirmation findings and complete GitHub, Zenodo, media and guarded public site readback.; active forecast 150 minutes (120-210); Fermi components Remaining assurance and archive: 1 x 40/50/70 minutes low/central/high (Bounded confirmation and deterministic repairs.); Page and communication assets: 1 x 40/50/70 minutes low/central/high (Established site generators.); CI, deployment and public readback: 1 x 40/50/70 minutes low/central/high (Two standard seal and deployment cycles.); positive-signal/closure probabilities 0.9/0.8 within 300 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 9/unknown/unknown/0/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 23; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -141 minutes; ratio 0.06; inside interval false; positive-signal/target-closure Brier scores 0.01/0.04; missing telemetry activeHumanMinutes: Human effort was not instrumented.; computeMinutes: Tool and compute execution were not separately metered.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Preserve the original one-run snapshot and record the two observed runs only in the terminal outcome. (reason: The intake snapshot counted the root run; the terminal response-receipt audit also includes the confirmation reviewer within the measured interval.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json
- Intended aims: science
- Artifact roles: research-output, evidence-assessment
- Decision object: counterexample — An observable-event contradiction to a specified limit-profile formula. Scope: Equal-half product-weight transpositions, fixed 0 < b < 1, s = log(6).
- Reusable methods: Structural compression (structural-compression); Explicit research-lineage reuse (research-lineage-reuse); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json
- Targeted clocks: assurance, publication
- Semantic bridge: explicit — Untouched slow labels imply fixed slow labels; event probability differences bound full total variation. Remaining risks: Written asymptotic argument awaits external scrutiny.; Priority search is bounded..
- Human judgement gates: Inspect the asymptotic moment argument and source correspondence.; Assess novelty and retain candidate qualifications.
- Next assurance action: Obtain external probability-specialist inspection and replay.
- Claim ceiling: Candidate counterexample to the pinned profile formula; no replacement profile or general sharp upper bound.
- Aim-scoped impact evidence:
  - science: NO_IMPACT_EVIDENCE — Inspectable candidate counterexample in Producer-coordinated publication; design none; comparator No matched comparator.; estimand No impact effect estimated.; no real-world effect evidence asserted
- Parent handoffs: depends-on-claim https://arxiv.org/abs/2409.16387v1; inherited claim: The stated equal-half profile conjecture and model conventions define the target.; inherited ceiling: The source conjecture is a target, not a proved input or external validation.



## Verification status

Unrefereed candidate. Internal model review and producer replay support inspection; formal verification, external reproduction, external peer review and priority remain unestablished. No replacement profile or general t_mix = t_star + O(n) upper bound is supplied.

## References

1. Nestoridi and Yan (2024), Cutoff for the biased random transposition shuffle, arXiv v1: Conjecture 1.6 is the precise target. <https://arxiv.org/abs/2409.16387v1>
2. Nestoridi and Yan (2025), FPSAC proceedings article: cutoff and spectrum results, distinct from the profile conjecture. <https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2025/6.pdf>
3. Nestoridi and Yan, official FPSAC 2025 poster: restates the profile conjecture. <https://www.math.sci.hokudai.ac.jp/sympo/fpsac2025/public/posters/107.pdf>
4. Teyssier (2020), Limit profile for random transpositions: the uniform-shuffle benchmark. <https://doi.org/10.1214/20-AOP1424>
5. Evidence Press (2026), Biased shuffles still cut off: related bounded-product cutoff candidate. <https://evidencepress.org/releases/bounded-product-transposition-cutoff/>