--- title: "Extremes do not always maximise: a minimal monotone reversible counterexample" date: 2026-08-29 version: "0.2.0-candidate" doi: 10.5281/zenodo.22163995 pdf: https://github.com/ipitchford/extremes-do-not-always-maximise/releases/download/v0.2.0-candidate/extremes-do-not-always-maximise-v0.2.0-candidate.pdf repository: https://github.com/ipitchford/extremes-do-not-always-maximise archive: https://zenodo.org/records/22163995 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Extremes do not always maximise: a minimal monotone reversible counterexample ## Plain-English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 $$ (1,\tfrac12,2,1). $$ 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 $$ P=\begin{pmatrix} 1/2&1/6&1/3&0\\ 1/3&1/3&0&1/3\\ 1/6&0&2/3&1/6\\ 0&1/6&1/3&1/2 \end{pmatrix}, $$ with stationary distribution `(2,1,4,2)/9` and spectrum `1,1/2,1/2,0`. For every integer $t\ge1$, $$ d_{\mathrm{TV}}\!\left(P^t(10,\cdot),P^t(01,\cdot)\right) =\frac23\left(\frac12\right)^{t-1}, $$ whereas $$ d_{\mathrm{TV}}\!\left(P^t(00,\cdot),P^t(11,\cdot)\right) =\left(\frac12\right)^t. $$ 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 $$ p=\frac{b}{1+b},\qquad r=\frac{c}{1+c},\qquad B=\frac{p+r}{2},\qquad A=1-B, $$ and $R=\max(p,r)$. Under the attractive condition $bc\le1$, the candidate derives three exact distance modes: $$ \text{extremal}=A^t,\qquad \text{incomparable}=RB^{t-1},\qquad \text{adjacent}=\frac{A^t+B^t}{2}. $$ 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: ```sh 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.sh ``` Then 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 1. **Zero-field theorem or counterexample.** Resolve the every-graph Ising question without extrapolating from the bounded four-vertex screen. 2. **Independent reconstruction.** Re-derive the kernel powers, family regimes and cardinality-minimality proof without importing the package formulas. 3. **Specialist prior-art audit.** Inspect the 1992 antecedent in full and search coefficient-of-ergodicity and binary Gibbs-sampler terminology. 4. **Formalisation.** Encode stochastic monotonicity, reversibility, row total variation and the three-state CDF argument in a proof assistant. 5. **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](https://github.com/ipitchford/extremes-do-not-always-maximise/releases/tag/v0.2.0-candidate) and [version DOI](https://doi.org/10.5281/zenodo.22163995) are the pinned inspection points. ## Searchable mathematical objects ### 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. ## Open directions for follow-up research - 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. ## 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:extremes-do-not-always-maximise - Attempt and metric receipts: ep-attempt:extremes-do-not-always-maximise-assurance-publication: published / positive; scope assurance-through-publication; target Reach internal Accept or PASS_WITH_NOTES on a source-faithful repaired candidate, then publish byte-identical immutable GitHub and Zenodo assets plus a guarded Evidence Press page whose canonical readback preserves the unrefereed claim ceiling.; active forecast 150 minutes (90-240); Fermi components review triage and closest-prior-art resolution: 1 x 30/45/75 minutes low/central/high (One pivotal 1992 full-text dependency, focused citation descendants and a finite response matrix.); package repair, rebuild, deterministic baseline and editorial gate: 1 x 20/35/60 minutes low/central/high (One six-page paper, a small exact verifier, targeted controls and one five-role round with at most one confirmation.); public research repository, CI, immutable release and Zenodo identity: 1 x 15/25/40 minutes low/central/high (One compact anonymous candidate with standing component licences and standard GitHub/Zenodo coupling.); Evidence Press authoring, media, composite seals, two deployments and readback: 1 x 25/45/65 minutes low/central/high (One new reader-first release using deterministic house art, audio, thumbnail, A/B then C/D protocol seals and existing guarded deployment tooling.); positive-signal/closure probabilities 0.85/0.7 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 95/unknown/0/6/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 7; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -55 minutes; ratio 0.63; inside interval true; positive-signal/target-closure Brier scores 0.0225/0.09; missing telemetry activeHumanMinutes: No instrument captured human direction or review time at the prospective attempt boundary.; deduplicatedModelTokens: The supported runtime goal counter began at 2026-08-29T16:00:08Z, 36 minutes after attempt registration. Its partial reading is retained in the closeout receipt, but a full-attempt fork-aware value cannot be recovered without reconstruction.; uncachedInputTokens: The runtime does not expose an uncached-input counter.. 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, method-demonstration, communication - Decision object: counterexample — A cardinality-minimal four-state monotone reversible heat-bath counterexample, together with an exact endpoint-symmetric two-spin family classification. Scope: The separately quantified general monotone-reversible clause of AIM problem 1.6(1); not the all-graph zero-field Ising question or diagnostics 1.6(2-3). - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); 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 — The AIM general clause is mapped to the Boolean-diamond state order, random-scan heat-bath kernel, total-variation row diameter and exact all-time formulas; the zero-field clause is separately excluded. Remaining risks: The closest 1992 coefficient-of-ergodicity article could not be inspected beyond its publisher abstract, bibliography and cited-by records.; No unaffiliated specialist has assessed the proof or source map.; The all-graph zero-field Ising question remains open.. - Human judgement gates: Assess the universal written proofs rather than inferring them from finite replay.; Assess whether the accessible-source novelty audit supports release at the bounded unrefereed-candidate ceiling.; Do not promote the bounded exact zero-field enumeration into an all-graph theorem.; Require a new release decision before claiming independent reproduction, formal verification, specialist review, peer review, novelty or priority. - Next assurance action: Obtain unaffiliated specialist review and lawful full-text comparison with Pflug and Schachermayer before any novelty or priority upgrade. - Claim ceiling: Unrefereed theorem candidate with producer-side exact replay, a separate producer cross-check and internal editorial review; not an all-graph zero-field result, independent reproduction, formal proof, specialist review, external peer review, novelty or priority claim. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — More reliable evaluation of extremal-start row-diameter claims in Finite monotone reversible Markov chains and two-spin heat-bath dynamics; design none; comparator No matched conventional research or publication workflow was prospectively registered.; estimand No effect on discovery time, error rate, review effort, theorem production, adoption or citation was estimated.; no real-world effect evidence asserted - Defeasible explanations: Certificate-first publication, structural compression and retained stop receipts reduce elapsed time and total human effort per reusable research result at a matched assurance boundary. (research-cycle-acceleration). IBE ledger: https://evidencepress.org/api/ibe-ledger.json ## 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. ## References 1. AIM Problem Lists. Markov chain mixing times, Problem 1.6. 2. Peres, Y., & Winkler, P. (2013). Can extra updates delay mixing? Communications in Mathematical Physics, 323, 1007-1016. 3. Fill, J. A., & Kahn, J. (2013). Comparison inequalities and fastest-mixing Markov chains. Annals of Applied Probability, 23, 1778-1816. 4. Pflug, G. C., & Schachermayer, W. (1992). A remark on the coefficient of ergodicity. Probability Theory and Related Fields, 93, 283-293. 5. Qin, Q. (2025). Analysis of two-component Gibbs samplers using the theory of two projections. Annals of Applied Probability, 35, 891-921.