Press release · 29 August 2026 · version 0.2.0-candidate
Extremes do not always maximise: a minimal monotone reversible counterexample
A four-state monotone reversible heat-bath chain is furthest apart at two incomparable states—not its minimum and maximum—at every positive time.
Plain-English summary
Imagine a process that moves among ordered states while respecting the order: starting higher should never make its future distribution stochastically lower. If the process also runs the same way in equilibrium forwards and backwards, it is natural to guess that the lowest and highest starting states are always the furthest apart.
This anonymous, unrefereed candidate shows that the guess is false in general. The example has four states arranged as a diamond: 00 at the bottom, 11 at the top, and the incomparable middle states 10 and 01. A random-scan single-site heat-bath update uses Gibbs weights
The resulting chain is irreducible, aperiodic, reversible and monotone. At every positive integer time, however, the middle pair is further apart in total variation than the bottom and top. The ratio is always exactly 4/3.
Four states are also minimal: with a unique bottom and top, every poset on at most three states is a chain, and monotonicity forces the extremal pair to realise the row diameter.
Exact result
In state order (00,10,01,11), the heat-bath kernel is
with stationary distribution (2,1,4,2)/9 and spectrum 1,1/2,1/2,0. For every integer $t\ge1$,
whereas
Each of the four adjacent pairs also has distance $(1/2)^t$. The incomparable pair is therefore the unique maximiser at every positive time.
Why the mechanism works
For the endpoint-symmetric family of weights (1,b,c,1), put
and $R=\max(p,r)$. Under the attractive condition $bc\le1$, the candidate derives three exact distance modes:
For (b,c)=(1/2,2), both eigenmodes have base 1/2, while the incomparable mode starts at 2/3. Its fixed leading advantage produces the all-time 4/3 gap. Other parameters can switch maximiser: the package includes an exact example where the middle pair leads through time 32 and the extrema lead from time 33.
What this does not settle
The AIM page places two questions close together. One asks about zero-field ferromagnetic Ising Glauber dynamics on every graph. The next, more general clause asks about any monotone reversible chain. This candidate negates the general clause only.
The four-state example has opposing site-dependent external fields. It is not a zero-field Ising example, and it says nothing decisive about all zero-field graphs. It also does not answer AIM diagnostics 1.6(2) or 1.6(3).
The bounded zero-field residue
The archive contains a separate exact screen over every connected labelled simple graph with two through four vertices, five temperatures 1, 3/2, 2, 3, 10, and times one through six. The canonical receipt covers:
- 43 graphs;
- 215 graph-temperature trajectories;
- 1,290 kernel powers; and
- 140,340 unordered row-pair comparisons.
The extremal pair attained every tested diameter. That result is useful as a regression fixture and a record of where no small counterexample was found. It is finite evidence only, not an all-graph theorem and not evidence that the zero-field statement is true.
Evidence and assurance boundary
The public source package supplies the six-page proof, accessible Markdown, exact rational verifiers, a separately structured producer cross-check, seven semantic negative controls, deterministic PDF rebuilding, a complete SHA-256 manifest, source-correspondence and novelty reports, and producer-organised five-role review with exact-byte confirmation.
A fresh extraction of the DOI-bearing ZIP passed all 18 release gates. The tagged GitHub workflow passed on Python 3.11 and 3.13, each in normal and optimized mode. GitHub and Zenodo copies of the PDF, ZIP and checksum ledger match the reviewed local assets byte for byte.
These facts establish availability, integrity, producer-side replay and the declared finite computations. They do not establish:
- independent rerun or independently authored reimplementation;
- proof-assistant formalisation;
- external Markov-chain specialist review;
- journal or comparable editorial peer review;
- exhaustive novelty or absolute priority; or
- resolution of the all-graph zero-field question.
The universal claims rest on the written proof. Successful code execution checks its formulas and examples but cannot replace mathematical assessment of its quantifiers.
Relationship to earlier work
The source is AIM Problem 1.6(1). Nearby work studies censoring from extremal configurations, comparison inequalities, two-component Gibbs samplers, order-adapted metrics and coefficients of ergodicity. A formula-first search found no matching four-state matrix, all-time distance sequences, 4/3 ratio or family classification.
That search has a material gap. The closest 1992 coefficient-of-ergodicity article by Pflug and Schachermayer was available only through its abstract, bibliography and cited-by metadata, not full text. The elementary example may also be known under different terminology or in informal circulation. Accordingly, the release makes no claim to be new, first or historically definitive.
Who should care
| Reader | What is useful now | Principal caution |
|---|---|---|
| Markov-chain researchers | A minimal exact stress test for extremal-start intuition | Check the proof and prior art independently |
| Gibbs-sampler researchers | A complete two-spin family with explicit switch regimes | The permanent witness uses external fields |
| Software authors | A four-state regression fixture for row-diameter assumptions | Passing the fixture does not validate a general algorithm |
| Formalisers | A small rational kernel with short universal formulas | No proof-assistant development is supplied |
| Research-methods readers | A package that separates theorem, bounded screen, replay and review | Internal role separation is not external validation |
How to inspect and reproduce it
Use tag v0.2.0-candidate or the Zenodo version DOI rather than the moving main branch. Python 3.11 or later is sufficient for the exact verifiers:
python3 verify_two_spin_family.py
python3 -O verify_two_spin_family.py
python3 crosscheck_formulas.py
python3 -O crosscheck_formulas.py
python3 test_negative_controls.py
python3 -O test_negative_controls.py
python3 verify_zero_field_bounded.py
python3 -O verify_zero_field_bounded.py
./build_paper.shThen inspect paper.md, CLAIMS.json, ASSURANCE.md, SOURCE_CORRESPONDENCE.md and NOVELTY_REPORT.md. The versioned manifest binds every release file.
Most valuable next projects
- Zero-field theorem or counterexample. Resolve the every-graph Ising question without extrapolating from the bounded four-vertex screen.
- Independent reconstruction. Re-derive the kernel powers, family regimes and cardinality-minimality proof without importing the package formulas.
- Specialist prior-art audit. Inspect the 1992 antecedent in full and search coefficient-of-ergodicity and binary Gibbs-sampler terminology.
- Formalisation. Encode stochastic monotonicity, reversibility, row total variation and the three-state CDF argument in a proof assistant.
- Remaining AIM diagnostics. Treat questions 1.6(2) and 1.6(3) as separate projects with their own quantifiers and assurance gates.
What is in the public package
The immutable release includes the manuscript PDF and sources, accessible Markdown, exact verifiers, mutation tests, claim and nonclaim registry, environment record, replay receipt, operating-model and metrics records, source and novelty audits, internal review reports, licensing files and a complete checksum manifest. The v0.2.0 candidate release and version DOI are the pinned inspection points.
Searchable mathematical objects
Exact text is provided alongside typeset mathematics so people and automated research tools can find and compare the objects without inferring a stronger assurance state.
Four-state all-time row-TV counterexample
- Exact searchable text
For weights (1,1/2,2,1), the incomparable row distance is (2/3)(1/2)^(t-1) and the extremal distance is (1/2)^t for every integer t at least 1.- LaTeX
d_{\mathrm{TV}}(P^t(10,\cdot),P^t(01,\cdot))=\frac23\left(\frac12\right)^{t-1}>\left(\frac12\right)^t=d_{\mathrm{TV}}(P^t(00,\cdot),P^t(11,\cdot))- Object type
- counterexample
- Relation to release
- claimed-result
- Scope
- The specified random-scan two-spin heat-bath chain; the scholarly object is an anonymous unrefereed candidate with producer-side replay, not an independently validated theorem.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- Determine whether the extremal pair always realises row-TV diameter for zero-field ferromagnetic Ising heat-bath dynamics on every finite graph.
- Obtain an unaffiliated Markov-chain specialist reconstruction of the four-state proof, family classification, source mapping and cardinality-minimality argument.
- Inspect Pflug and Schachermayer's 1992 coefficient-of-ergodicity article lawfully in full and conduct a broader specialist novelty and priority audit under alternative terminology.
- Formalise the Boolean-diamond kernel, stochastic monotonicity, exact powers and three-state CDF argument in a proof assistant.
- Address AIM Problem 1.6 diagnostics (2) and (3), which are not answered by this counterexample.
Verification status
Anonymous, unrefereed theorem candidate at internal PASS_WITH_NOTES. The universal counterexample, family classification and cardinality-minimality claims rest on the written proofs; the programs replay exact formulas and stated finite predicates. The closest 1992 coefficient-of-ergodicity paper could not be inspected in full, so the bounded prior-art search does not establish novelty or priority. No unaffiliated rerun, independent reimplementation, proof-assistant formalisation, external specialist review or editorial peer review has occurred. The zero-field Ising question on every graph and AIM diagnostics 1.6(2) and 1.6(3) remain open.
Cite
BibTeX
@misc{extremesdonotalwaysmaximise2026,
title = {Extremes do not always maximise: a minimal monotone reversible counterexample},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22163995},
url = {https://doi.org/10.5281/zenodo.22163995},
version = {0.2.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/extremes-do-not-always-maximise/}
}Also: cite.bib · paper.json · this page as Markdown