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:

        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 $a<r$, $b<r$ and $c<b$. With birational boundary labels fixed to one and toggles applied from top to bottom,

If $t>1$ is the positive root of $t^9-t-1$, then

is fixed. Writing $p=1/(1+t)$, the logarithmic derivative has characteristic polynomial

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

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:

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

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:

python3 verify_theorem.py
python3 -O verify_theorem.py
node verify_theorem.mjs
python3 test_mutations.py

For the complete manifest and document gates:

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.

@misc{fourelementrowmotionperiodicity2026,
  title        = {A four-element obstruction to birational and piecewise-linear rowmotion periodicity},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22178666},
  url          = {https://doi.org/10.5281/zenodo.22178666},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/four-element-rowmotion-periodicity/}
}