---
title: "Exact profiles and a vanishing-bias crossover for two-class random transpositions"
date: 2026-09-12
version: "0.1.0-candidate"
doi: 10.5281/zenodo.22727156
pdf: https://github.com/ipitchford/two-class-transposition-profiles/releases/download/v0.1.0-candidate/two-class-transposition-profiles-0.1.0-candidate.pdf
repository: https://github.com/ipitchford/two-class-transposition-profiles
archive: https://zenodo.org/records/22727156
license: CC0-1.0
status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)
---

# Exact profiles and a vanishing-bias crossover for two-class random transpositions

## Summary

If half the cards in a shuffle are chosen less often than the other half, what remains of the initial order just as the deck approaches randomness? This unrefereed candidate gives an exact limiting answer for the whole permutation, not merely a lower bound from one statistic. Two counts—fixed cards in the slow and fast classes—capture the remaining discrepancy asymptotically.

It also describes how the answer changes as the bias disappears. The transition occurs when the bias is of order one over the logarithm of the deck size. The proof is written mathematics; finite computer checks corroborate particular identities, not the infinite-size theorem.

## Summary for specialists

Let $N=2n$, with two classes of size $n$. Independently sample two labels with probabilities $b/N$ in the slow class and $(2-b)/N$ in the fast class, then transpose them; repeated samples give the identity. Start from the identity permutation. At $t_N(s)=\lfloor N(\log N+s)/(2b)\rfloor$, the candidate proves stationary-$L^2$ approximation of the full density by the normalized tilt $(1+r_A)^{F_A}(1+r_B)^{F_B}$.

For fixed $0<b<1$, $(r_A,r_B)=(e^{-s},0)$. For $b_N\to1$ with $(1-b_N)\log N\to\lambda\in[0,\infty]$, the pair is $(e^{-s},e^{-s-2\lambda})$, with the second coordinate zero at infinity. Consequently the limiting full total-variation distance is

$$
\left\|\operatorname{Pois}\!\left(\frac{1+r_A}{2}\right)\otimes\operatorname{Pois}\!\left(\frac{1+r_B}{2}\right)-\operatorname{Pois}(1/2)^{\otimes2}\right\|_{\rm TV}.
$$

Convergence is locally uniform in $s$. No uniform statement as $b\downarrow0$, unequal-class theorem or arbitrary-weight extension is claimed.

## Technical account

The proof has three distinct steps. First, a trace comparison between several tracked labels and independent single-label walks gives all fixed-degree mixed factorial moments. Collision terms are bounded in operator norm, avoiding a multiplying factor from the rapidly growing state space.

Second, the full squared density norm is computed through the representation spectrum. The fixed-bias proof imports explicitly identified bulk estimates from Nestoridi–Yan. A separate uniform estimate handles vanishing bias; fixed-parameter constants are not silently made uniform. Stable Littlewood–Richardson identities identify the bounded-degree contribution, and direct suppression handles the infinite crossover endpoint.

Third, the fixed-point tilt has the same limiting squared norm as the complete shuffle density. Positivity of its factorial expansion and Cauchy–Schwarz force their $L^2$ distance to vanish. This last step is the bridge from observables to the whole law: Poisson count limits alone would not suffice.

## Evidence, assurance and limitations

The package contains a complete written candidate proof, exact finite checks, numerical diagnostics, source locators and an internal editorial response. Fresh ordinary and optimized Python runs pass 112 exact coloured-moment checks; deliberately corrupted self-loop probabilities are rejected. A supplied spectral checker was inspected and rerun, passing 104 trace comparisons and 358 stable restriction identities within its stated finite scope.

Internal AI review and supplied review reports do not establish unaffiliated specialist validation or external journal peer review. Formal verification and historical priority are not established. Finite-size permutations can retain information beyond the two counts; the theorem is asymptotic. The model uses label weights, two equal classes, independent sampling with replacement and the stated discrete clock.

## Relationship to earlier work

The earlier Evidence Press counterexample ruled out a proposed total-fixed-point profile using a slow-class event. This successor claims the replacement full-law profile and its crossover, rather than reclassifying the earlier lower bound as a full solution. The parent remains an immutable, separately scoped research object.

Nestoridi–Yan provide the model spectrum and fixed-bias bulk input. Teyssier's uniform-shuffle theorem is recovered at the zero-crossover endpoint. His 2026 conjugacy-invariant framework explains a related finite-level character mechanism but does not directly identify this noncentral coloured density. Jain–Sawhney's planted fixed-set approximation motivates an alternate route, not a dependency of the final argument. These comparisons are bounded source positioning, not priority clearance.

## Who should care, and why

| Audience | Potential use | Required caution |
|---|---|---|
| Mixing-time researchers | Inspect an exact profile and a vanishing-bias scale. | Check the full-norm argument and pinned source dependencies. |
| Representation theorists | Reuse the bounded-degree restriction calculation. | Equal-class and fixed-degree assumptions matter. |
| Research agents | Replay finite checks and follow the claim/source map. | Replay and DOI availability do not certify the theorem. |
| Interested readers | See why counting all fixed cards can hide class information. | The asymptotic conclusion is not exact finite-deck sufficiency. |

## Why the problem matters

Cutoff locates the rapid transition to randomness; a profile identifies its shape. Biased sampling introduces a second question: how small must the bias become before the uniform answer returns? This candidate links that scale to an explicit two-coordinate limiting law and makes the claimed residual information precise. No practical shuffling standard, security guarantee or measured research-productivity benefit follows from it.

## How to inspect or reproduce the recorded checks

Download and extract the versioned ZIP. First run `python3 code/package.py --verify`, then `python3 code/independent_check.py` and its `python3 -O` variant. Each ordinary run should report PASS and 112 coloured-moment checks. The `--negative-control` variant must fail with an incorrect-self-loop message. These exact checks use only the Python standard library; the README gives separate pinned numerical and PDF-build instructions.

## The most valuable next projects

- Extend the density argument to unequal or additional sampling classes, with new spectral-tail control.
- Investigate bias tending to zero, which lies outside the current uniform estimates.
- Quantify finite-size errors in conditional sufficiency instead of treating the limiting statement as a finite certificate.
- Test whether the positive-expansion and norm-matching argument applies to other noncentral walks.

## What is in the evidence package

The versioned archive includes the formatted paper and Markdown/TeX source, claim index, current disposition, preserved historical audits, source-number crosswalk, replay instructions and receipts, original code, internal editorial reports, complete file manifest and component licences. Original prose/data use CC0 and original code uses MIT. Third-party source texts and unlicensed supplied-review files are not silently redistributed. GitHub and Zenodo carry byte-matched release assets.




## Open directions for follow-up research

- Unequal or additional sampling classes require new estimates.
- Bias tending to zero lies outside the uniform theorem.
- Quantitative finite-size conditional-sufficiency errors remain open.
- Transfer the norm-saturation mechanism to other noncentral walks.

## 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:two-class-transposition-profiles
- Attempt and metric receipts: ep-attempt:two-class-transposition-profiles-assurance-publication: published / positive; scope assurance-through-publication; target Complete review revisions and publish GitHub/Zenodo assets and guarded Evidence Press page, media and readback.; active forecast 90 minutes (60-150); Fermi components Source repairs and internal editorial gate: 1 x 20/30/50 minutes low/central/high (Established publication route; not an empirical speed comparison.); Immutable archives and communication assets: 1 x 20/30/50 minutes low/central/high (Established publication route; not an empirical speed comparison.); Composite CI and two deployment cycles: 1 x 20/30/50 minutes low/central/high (Established publication route; not an empirical speed comparison.); positive-signal/closure probabilities 0.95/0.85 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 18/unknown/unknown/0/14/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -72 minutes; ratio 0.2; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: Human effort was not instrumented.; computeMinutes: Substantive computation time was not separately instrumented.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Opening 1 retained; terminal 6. (reason: Opening root-only snapshot retained; terminal count includes five completed editorial agents.). 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, communication
- Decision object: bound — Full-density and total-variation profile in two bias regimes. Scope: Two equal sampling classes, identity start, independent label samples with replacement.
- Reusable methods: Structural compression (structural-compression); Exact regime stitching (regime-stitching); 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 — Coloured moments, full spectral norm and positive norm saturation identify the complete permutation density. Remaining risks: Written proof and imported source estimates require mathematical judgment.; Finite checks cannot certify all-degree asymptotic estimates.; Bounded source search does not establish priority..
- Human judgement gates: Assess the full-law bridge and imported estimates.; Preserve scope and historical source boundaries.; Separate mathematical validity, novelty and publication.
- Next assurance action: Inspect the written full-law argument and test extensions as separate research; external review is not a publication prerequisite.
- Claim ceiling: Unrefereed equal-class, identity-start full-density and TV profile candidate. Finite checks do not certify asymptotics. No uniformity as b tends to zero, unequal-class extension, external validation, formal verification, priority or impact claim.
- Aim-scoped impact evidence:
  - science: NO_IMPACT_EVIDENCE — Inspectable two-class profile and crossover candidate in Producer-coordinated mathematical publication; design none; comparator None.; estimand No acceleration or impact effect estimated.; no real-world effect evidence asserted
- Parent handoffs: extends-result ep-work:biased-transposition-profile-counterexample; inherited claim: Earlier slow-subset profile obstruction is reconstructed and extended to a full-law profile.; inherited ceiling: Unrefereed parent; reuse is not independent confirmation.



## Verification status

Unrefereed equal-class, identity-start full-density and TV profile candidate. Finite checks do not certify asymptotics. No uniformity as b tends to zero, unequal-class extension, external validation, formal verification, priority or impact claim.

## References

1. Nestoridi and Yan, Cutoff for the Biased Random Transposition Shuffle, arXiv:2409.16387v1: spectrum and fixed-bias bulk input. <https://arxiv.org/abs/2409.16387v1>
2. Anonymous, Untouched subsets and a counterexample to the biased-transposition limit-profile conjecture, Evidence Press (2026): antecedent lower-bound obstruction. <https://doi.org/10.5281/zenodo.22708622>
3. Teyssier, Limit profile for random transpositions (2020): uniform endpoint. <https://doi.org/10.1214/20-AOP1424>
4. Teyssier, Cutoff profiles for conjugacy invariant random walks on symmetric groups, arXiv:2605.28770v2 (2026): related central-character framework. <https://arxiv.org/abs/2605.28770v2>
5. Jain and Sawhney, Hitting time mixing for the random transposition walk (2026): planted fixed-set comparison. <https://doi.org/10.1007/s00440-026-01479-9>
6. Francis and Nestoridi, Limit Profiles for Separation Distance, arXiv:2605.19084v1 (2026): different metric and singleton-class perturbation. <https://arxiv.org/abs/2605.19084v1>