A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture

Summary

External development, 12 September 2026: an unrefereed preprint by liqsweep proposes a general proof of polynomial-composition rigidity. Conditional on the external rigidity proof and Furter's Theorem B, every length-two closure equality follows, including this e=3 containment and all fixed-e singleton containments. That alternative argument does not validate this release's Fourier estimates, interval certificates or smooth-point construction. Internal source review and scope audit are not unaffiliated verification. Original archives and release-time audio/video are unchanged.

The setting. A plane polynomial automorphism is an invertible map of the plane whose formula, and whose inverse, are both given by polynomials. Every such map can be built by composing simple "elementary" maps, and the degrees of those building blocks form its polydegree. Grouping maps by polydegree gives families, and a natural question is which families can be approached as limits of others — whether one family sits inside the closure of another.

The question. The Polydegree Conjecture asks when one such family lies in the closure of another. A long-standing column of that problem asks whether

holds for every integer $d\ge2$.

In words: every map of polydegree $(d+3)$ arises as a limit of maps of polydegree $(d,4)$, for every $d\ge2$. Here $e=3$ is the fixed offset in this family of containments, and the "column" is the whole list of cases as $d$ varies.

What was found. This anonymous, unrefereed candidate proves that full $e=3$ column. It first turns the Lewis--Perry--Straub determinant criterion into a local geometric question: find one smooth common zero of two coefficient polynomials where the next coefficient does not vanish. A Fourier-exponential limit supplies transverse model zeros for the two nontrivial residue classes. Quantitative Newton--Kantorovich bounds then carry those zeros to all sufficiently large degrees, while exact finite-field and interval certificates close the remaining ranges.

The archived finite bridge contains 97,033 ordered FLINT/Arb cases with no failure. A separate exact-rational envelope handles every later degree. The package includes the formulas, normative checker specification, locators, complete case rows, receipts, source code, manifests and review dispositions needed to inspect that claim.

Candidate status: anonymous · unrefereed · producer-side theoretical and computer-assisted proof · no unaffiliated full Arb rerun, separately authored checker, proof-assistant formalisation, external specialist review or editorial peer review.

Summary for specialists

Lewis, Perry and Straub give a sufficient criterion for $\mathcal G_{(d+e)}\subseteq\overline{\mathcal G_{(d,e+1)}}$ using the coefficient polynomials $g_{n,e}$ and an auxiliary determinant $a_{d,e}$. The candidate proves

and an integral weighted-Euler syzygy which, on $V(g_{d,e},\ldots,g_{d+e-2,e})$, factors this determinant through a single affine Jacobian minor and $g_{d+e-1,e}$. The criterion is therefore equivalent to a smooth point of codimension $e-1$ off the next hypersurface. The corresponding Hensel certificate uses an $(e-1)\times(e-1)$ minor and needs no exclusion of primes dividing $d+e-1$.

For $e=3$, write $d=3m+r$. The $r=1$ class has the exact special point $(0,0,1)$. In the other two classes, factorially normalised coefficient polynomials converge in $C^1$ on compact subsets of $\mathbf C^2$ to explicit root-of-unity Fourier sums

A discrete-Fourier concentration constructs transverse zeros of the relevant pairs. An exact-rational $C^1$ defect envelope, a stated complex sup norm, and a quantitative Newton--Kantorovich argument give persistence for $m\ge48{,}550$. The intermediate range $33\le m<48{,}550$ in residues $0$ and $2$, except the already covered pair $(33,0)$, is certified case by case with outward-rounded complex balls. Exact finite-field checks cover $2\le d\le100$.

Technical account

The proof is a gap-free stitch of four regimes.

The normative certificate specification binds each printed mathematical predicate to the implementation. It defines the non-zero row scaling, exact coefficient table, complex sup norm, induced row-sum matrix norm, Gaussian-rational locators, cutoff and tail majorants, residual and inverse bounds, Neumann correction, Newton radius, third-row non-vanishing and transported Jacobian margin. Acceptance is fail-closed and does not depend on Python assert.

The finite receipt is required to contain exactly $2(48{,}550-33)-1=97{,}033$ cases in the prescribed order, no failures and the terminal CERTIFIED state. Embedded SHA-256 digests bind it to the checker, locator sequence and complete row file. The eventual receipt is produced from exact rational arithmetic. A fresh producer replay under pinned python-flint is recorded separately from the historical archive receipt.

The incoming final review classified the paper as minor revisions. The release incorporates its requested normative formula-to-checker bridge, explicit complex norm, fuller $C^1$ derivative derivation, printed tail ratios, regime window, row-scaling explanation, exact case count, claim-to-evidence map and calibrated assurance language. No fatal or major mathematical defect was reported.

What is classical, and what is offered here

The Polydegree Conjecture, coefficient polynomials and sufficient specialisation criterion come from Lewis, Perry and Straub. Edo's theorem supplies the congruence class $d\equiv1\pmod3$, and Perry's dissertation is a direct antecedent to the computational programme. A preceding Evidence Press candidate established the Jacobian--Euler interpretation and smaller Hensel certificate while leaving the full $e=3$ column open.

This candidate's main increment is the all-degree closure: the explicit Fourier limit, transverse limiting zeros, quantitative persistence argument, exact eventual threshold and complete finite bridge, combined with the inherited smooth-point reduction in one proof. Its fixed-$e$ continuation is a research programme and conjecture, not a theorem of this release.

The documented public-record search, frozen on 12 August 2026, found no earlier published proof of the full $e=3$ column. Under the project's published-record novelty criterion, an earlier published antecedent would falsify that novelty statement. The search is dated and bounded; it is not a specialist priority adjudication.

Evidence, assurance and limitations

Established within the package.

• Exact algebraic and finite-field checks replay in ordinary and optimized Python.
• An exact-rational positive envelope proves the eventual branch from $m=48{,}550$.
• A pinned Python-FLINT/Arb checker certifies 97,033 ordered finite cases with no failures.
• Locator, row, checker, receipt, PDF and package identity are bound by SHA-256 manifests, and the formula-to-checker bridge is stated normatively in CERTIFICATE_SPEC.md.
• The all-degree range partition has no uncovered degree.

Review and replay. Review consisted of internal and cross-model adversarial review and a final minor-revisions review; there has been no specialist peer review. The final reviewer could not run Python-FLINT in that review environment, and instead audited the checker source and reconciled every archived row, order, count, hash and reported extremum. A later producer-side fresh replay in the pinned environment (12 August 2026) regenerated rows and receipts byte-identical to the reviewed archive. Neither action is unaffiliated reproduction.

Known weak points. The package's review guide asks readers to start with: the formula-to-checker bridge (the analytic derivation of the tails and the row-scaling claim); the Kantorovich conventions on $\mathbf C^2$ with the $\ell^\infty$ norm; sign bookkeeping in the general-chart syzygy; the coverage of the three-regime split in the $C^1$ assembly constants; making sure nothing load-bearing rests on a grid-based consistency check; the absence of independent reconstruction; and the coverage of the novelty search.

What the result does not establish

• It does not prove the full Polydegree Conjecture or the general fixed-$e$ extension.
• It does not show that the supplied checker is independently implemented or that the proof has been independently reconstructed.
• It does not establish formal verification, external specialist review, editorial peer review or venue acceptance.
• It does not turn cross-model producer review, exact replay, CI, hashes, GitHub, Zenodo or Evidence Press publication into independent mathematical assurance.
• It does not establish research-workflow acceleration or impact; no prospective matched comparator or complete research clock was recorded.

Who should care, and why

Specialist audience candidates

Direct specialist audiences include researchers in polynomial automorphism groups, affine algebraic geometry, multivariate Hensel lifting, special-function asymptotics, validated numerics and computer-assisted proof. The most informative review would combine structural checking of the Polydegree criterion with a separately authored interval implementation; either alone leaves a distinct bridge untested.

Why the problem matters

The Polydegree Conjecture asks how the families of plane automorphisms, sorted by polydegree, fit together under limits. Individual containments can be checked degree by degree with certificates, but a column such as $e=3$ contains infinitely many cases, so closing it needs an argument that works for all $d$ at once. This candidate supplies one: a smooth-point reformulation of the Lewis--Perry--Straub criterion, a Fourier-exponential limit for large degrees, and certified computation for the finite remainder. Those ingredients, rather than the single column alone, are what a reader might reuse; the fixed-$e$ continuation remains a research programme, and the external rigidity preprint noted above offers a separate, conditional route to the same containments.

How to inspect or reproduce the recorded checks

1. Start with the claim map. REVIEW_GUIDE.md lists the eight load-bearing claims, where each is stated and proved in paper/MANUSCRIPT.md (and the PDF), and which script validates it. CERTIFICATE_SPEC.md is the normative bridge from printed predicates to code.
2. Run the exact and consistency checks from verification/: python3 verify_g1a.py (about 11 s; expect ALL CHECKS PASS), python3 check_b1_constants.py, python3 probe_g1b.py and python3 check_b3.py (a few minutes each).
3. Run the rigorous branches in a fresh environment synced from requirements-certify.lock. The release-safe single command is

```sh CERT_PYTHON=.venv-certify/bin/python ./run_rigorous.sh

```

It runs the exact-rational eventual checker and the Arb positive and semantic negative controls in ordinary and optimized Python, then the full finite range. The recorded terminal result is ARB CERTIFICATE: 97033 certified, 0 failed followed by ATOMIC RIGOROUS REPLAY PASS.

1. Compare receipts. The finite receipt should record status = CERTIFIED, expected = certified = 97033 and failed = []; the eventual receipt records the endpoint bound 0.0029969535913992734 < 0.003 at m = 48550. REPLAY_RECEIPT.md lists the SHA-256 digests of the checker, locators, rows and receipts.

Matching hashes show that the archived checks reproduce; they do not test a separately transcribed specification or establish mathematical truth on their own.

The most valuable next projects

1. Independently reconstruct the coefficient normalisation, Fourier limit, Newton--Kantorovich bounds and Lewis--Perry--Straub implication from the definitions.
2. Implement the finite and eventual checkers independently and rerun the immutable public locator sequence without producer intermediates.
3. Obtain specialist reviews in polynomial automorphisms, algebraic geometry, asymptotic analysis and interval certification, retaining any corrections publicly.
4. Formalise the Jacobian--Euler factorisation and the analytic persistence theorem over an explicit trusted base.
5. Assess the external general rigidity proof before commissioning another fixed-$e$ containment proof. Extend Fourier persistence only for additional structural or certificate information; keep failures and changed thresholds visible.

What is in the evidence package

The public package contains the 13-page manuscript PDF and accessible Markdown source; the normative certificate specification; exact finite-field, symbolic and rational verification programs; the pinned FLINT/Arb checker; 97,033 complete case rows and their exact Gaussian-rational locators; finite and eventual terminal receipts; replay instructions and receipt; machine-readable claims, status and assurance records; review and revision dispositions; literature-search records; licences; provenance; CI; and SHA-256 manifests.

The Evidence Press page, cover image, Open Graph card, transcript and synthetic-voice audio briefing are communication surfaces. They are not additional mathematical evidence.

@misc{fulle3columnpolydegreeconjecture2026,
  title        = {A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.21909085},
  url          = {https://doi.org/10.5281/zenodo.21909085},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/full-e3-column-polydegree-conjecture/}
}