--- title: "A four-element obstruction to birational and piecewise-linear rowmotion periodicity" date: 2026-08-30 version: "0.1.0-candidate" doi: 10.5281/zenodo.22178666 pdf: https://github.com/ipitchford/four-element-rowmotion-candidate/releases/download/v0.1.0-candidate/four-element-rowmotion-paper.pdf repository: https://github.com/ipitchford/four-element-rowmotion-candidate archive: https://zenodo.org/records/22178666 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A four-element obstruction to birational and piecewise-linear rowmotion periodicity ## Summary Rowmotion repeatedly updates labels on a partially ordered set. Many important families are periodic: after finitely many updates, every generic labelling returns to where it began. This anonymous, unrefereed candidate gives a smallest possible obstruction for two standard lifted forms of rowmotion. The poset has four elements: ```text r / \ a b | c ``` Both birational rowmotion and order-polytope piecewise-linear rowmotion have infinite order on this poset. Exact exhaustive enumeration proves that no finite poset with three or fewer elements can have either behaviour. ## Summary for specialists Let $P=\{r,a,b,c\}$ have covers $a1$ is the positive root of $t^9-t-1$, then $$ x_t=(t^6,t^3,t^4,t^2) $$ is fixed. Writing $p=1/(1+t)$, the logarithmic derivative has characteristic polynomial $$ \lambda^4+2\lambda^3+(2+p)\lambda^2+2\lambda+1. $$ Finite order would put $p$ in a totally real cyclotomic subfield. But $\mathbb Q(p)=\mathbb Q(t)$, while $t^9-t-1$ is irreducible and has one real root and eight nonreal roots. This contradiction proves infinite birational order. For piecewise-linear rowmotion, the rational interior fixed point $$ y_*=\left(\frac34,\frac38,\frac12,\frac14\right) $$ lies in a strict linearity chamber. The local derivative there has a nontrivial Jordan block at eigenvalue $-1$. A finite-order map cannot have such a derivative at a fixed point. ## Technical account: two local obstructions The birational proof does not try to follow a generic orbit. It uses a fixed point to turn a global periodicity claim into a local linear-algebra test. If $R^N$ were the identity rational map, differentiating at the positive fixed point would give $J^N=I$. Every eigenvalue would be an $N$th root of unity, so every real coefficient of the characteristic polynomial would lie in a totally real cyclotomic field. The exact field signature rules this out. The piecewise-linear argument follows the same compression principle in a different category. At an interior point where the relevant maxima and minima are strict, the map is genuinely linear nearby. A nontrivial Jordan block survives every even power, so no iterate can be the identity on that neighbourhood. These are separate proofs. The piecewise-linear claim is not inferred from the birational obstruction. ## Why four elements are necessary Minimality is global over finite posets, not merely over rooted trees or fences. The verifier enumerates every labelled strict partial order through size three and then quotients by relabelling: | Size | Labelled strict orders | Isomorphism classes | Exact birational periods | |---:|---:|---:|---| | 0 | 1 | 1 | $1$ | | 1 | 1 | 1 | $2$ | | 2 | 3 | 2 | $2,3$ | | 3 | 19 | 5 | $2,4,6,6,6$ | All nine isomorphism classes therefore have finite birational order. Because these identities are subtraction-free, the standard tropicalisation principle transfers them to the corresponding piecewise-linear maps. Hence no poset on at most three elements is an obstruction for either lift. ## Evidence, assurance and limitations The immutable package contains the five-page DOI-bearing paper, aligned Markdown and proof certificate, exact Python replay, language-separated JavaScript corroboration, the exhaustive small-poset classification, five semantic mutations, source and convention correspondence, citation and bounded novelty audits, the supplied review and response, licences, environment declaration, checksum ledger and complete manifest. Public Linux CI verifies the shipped bytes, reruns Python normally and under optimization, rejects all five corrupted certificates, runs the JavaScript checks, rebuilds the PDF and inspects its structure. GitHub and Zenodo expose the same five release assets, and unauthenticated downloads matched the local files byte for byte. These checks establish availability, integrity and producer replay. They do not establish unaffiliated rerun or reimplementation, proof-assistant formalization, authenticated external specialist review, editorial peer review, historical priority or research impact. ## Relationship to earlier work The four-element Hasse shape is not claimed as new. Grinberg and Roby draw the same non-skeletal rooted tree, up to order duality, in their Example 70. Their nearby theorem proves finite order for skeletal posets and does not determine the order of this non-skeletal example. The 2015 AIM problem list asks broadly for the study of birational rowmotion on Proctor's d-complete posets. Dangwal and coauthors later reported experimental evidence for nonperiodicity on non-graded rooted trees without a proof. Up to reversal or duality, the present poset is the asymmetric two-segment fence $\breve F(2,3)$; Mertin and Poznanović provide the closest family-level context but not the two fixed-point obstructions or global four-element minimality proved here. The bounded search found no exact theorem collision. That is not proof of historical novelty, so the release makes no first or priority claim. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Algebraic combinatorialists | A sharp local obstruction on the smallest possible poset | Do not read it as a classification of d-complete posets | | Dynamical-systems researchers | A fixed-point derivative route from periodicity to field signature or Jordan form | The arguments depend on the stated toggle and boundary conventions | | Computational reviewers | A nine-class exact minimality fixture with hostile mutations | Reimplement independently rather than importing producer code | | Formalizers | Two compact local arguments plus a finite exhaustive base case | The rational-map domain and tropical transfer still need formalization | | Interested readers | A concrete example where a four-point order already supports nonperiodic lifted dynamics | Candidate publication is not field consensus | ## Why the problem matters Rowmotion links posets, dynamical algebraic combinatorics and representation- theoretic structure. Periodicity theorems are often signatures of hidden regularity. A size-minimal obstruction identifies exactly how little non-graded structure is needed for that regularity to fail, while the two different derivative arguments give reusable tests for other families. ## How to inspect or reproduce the checks Use tag `v0.1.0-candidate` or the version DOI, not moving `main`: ```sh python3 verify_theorem.py python3 -O verify_theorem.py node verify_theorem.mjs python3 test_mutations.py ``` For the complete manifest and document gates: ```sh bash run_all.sh ``` Expected markers include `PASS_EXHAUSTIVE_SMALL_POSET_MINIMALITY`, `LANGUAGE_SEPARATED_ALL_CHECKS_PASS`, `MUTATIONS_PASS 5/5` and `PACKAGE_PASS`. Successful execution confirms the encoded identities and finite enumeration. It does not establish novelty, independence or peer review. ## The most valuable next projects 1. Classify finite- and infinite-order birational rowmotion on non-graded rooted trees or more general d-complete posets. 2. Determine the periodicity boundary for asymmetric two-segment fences. 3. Reconstruct both local obstructions in a materially separate stack. 4. Formalize the field, Jordan and minimality arguments in a proof assistant. 5. Obtain an authenticated specialist review and wider priority audit. ## What is in the evidence package The all-files ZIP includes the DOI-bearing PDF and source, aligned Markdown, proof and theorem certificates, exact verifiers, exhaustive minimality output, tests and mutations, source/citation/novelty records, the supplied review and response, licences, runtime declaration, release notes, receipts, checksums and complete manifest. The version DOI is the citation target; any correction should be a versioned successor rather than a silent edit. ## Open directions for follow-up research - Classify finite-order and infinite-order birational rowmotion on non-graded rooted trees or on d-complete posets more broadly. - Determine which asymmetric two-segment fences have finite-order birational or piecewise-linear rowmotion. - Independently reconstruct both fixed-point derivative obstructions in a materially separate stack. - Formalize the birational field argument, piecewise-linear Jordan obstruction and exhaustive size-three minimality in a proof assistant. - Obtain an authenticated specialist review and a broader historical-priority assessment. ## 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:four-element-rowmotion-periodicity - Attempt and metric receipts: ep-attempt:four-element-rowmotion-periodicity-assurance-publication: published / positive; scope publication-only; target Publish the already-reviewed theorem candidate through a reader-first Evidence Press page with provenance-bound media, append-only operating records, composite A/B and C/D seals, zero-cost guarded deployment and exact canonical readback without raising its assurance or novelty ceiling.; active forecast 90 minutes (60-150); Fermi components Reader-first page, operating records and source-bound metadata: 1 x 15/25/40 minutes low/central/high (One five-page theorem candidate with established site schemas, explicit source ancestry and a multidimensional assurance ceiling.); Deterministic art, provenance-bound TTS audio, Open Graph image and thumbnail: 1 x 10/15/25 minutes low/central/high (One release using existing media generators and the mandatory OpenAI fable TTS path.); Composite content and ledger seals, public CI and merge: 1 x 15/25/40 minutes low/central/high (One isolated feature branch using the established content-then-ledger A/B pattern and current three-version Node matrix.); Zero-cost guarded deployment, canonical readback and C/D closeout: 1 x 20/25/45 minutes low/central/high (One new canonical URL requiring unavoidable public readback, publication-ledger append, second composite seal and final convergence check.); positive-signal/closure probabilities 0.98/0.9 within 150 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 30/unknown/3/3/0/8; cycles positive/negative/inconclusive 0/0/0; falsification gates 0; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -60 minutes; ratio 0.333; inside interval false; positive-signal/target-closure Brier scores 0.0004/0.01; missing telemetry activeHumanMinutes: No instrument captured human direction or review time at the prospective publication-only boundary.; deduplicatedModelTokens: The runtime did not expose a fork-aware task-local model-token counter for this publication-only attempt.; uncachedInputTokens: The runtime does not expose an uncached-input token 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: obstruction — Two exact local derivative obstructions proving infinite order for the birational and piecewise-linear lifts on one four-element poset, together with exhaustive global size-minimality. Scope: The poset with covers a 2. Grinberg, D., & Roby, T. (2016). Iterative properties of birational rowmotion I: Generalities and skeletal posets. Electronic Journal of Combinatorics 23(1), P1.33. 3. Einstein, D., & Propp, J. (2014). Piecewise-linear and birational toggling. DMTCS Proceedings, AT, 513–524. 4. Dangwal, P., Kimble, J., Liang, J., Lou, J., Sagan, B. E., & Stewart, Z. (2023). Rowmotion on rooted trees. Séminaire Lotharingien de Combinatoire 88, B88a. 5. Proctor, R. A., & Scoppetta, L. M. (2019). d-Complete posets: Local structural axioms, properties, and equivalent definitions. Order 36, 399–422. 6. Mertin, A., & Poznanović, S. (2024). Toggleability spaces of fences. Electronic Journal of Combinatorics 31(4), P4.46.