--- title: "A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture" date: 2026-08-12 version: "0.1.0-candidate" doi: 10.5281/zenodo.21909085 pdf: https://github.com/ipitchford/polydegree-full-e3-column/releases/download/v0.1.0-candidate/polydegree-full-e3-column-v0.1.0-candidate.pdf repository: https://github.com/ipitchford/polydegree-full-e3-column archive: https://zenodo.org/records/21909085 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A Jacobian smooth-point criterion and the full e=3 column of the Polydegree Conjecture ## Summary Plane polynomial automorphisms can be sorted by the degree sequence of the elementary maps used to build them. The Polydegree Conjecture asks when one such family lies in the closure of another. A long-standing column of that problem asks whether $$ \mathcal G_{(d+3)}\subseteq\overline{\mathcal G_{(d,4)}} $$ holds for every integer $d\ge2$. 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 $$ a_{d,e}=(-1)^e\det D(g_{d,e},\ldots,g_{d+e-1,e}) $$ 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 $$ H_q(\mathbf X)=\frac13\sum_{j=0}^2\omega^{-qj} \exp\!\left(\rho^2\omega^jX_1+\rho\omega^{2j}X_2\right), \qquad \rho^3=-4. $$ 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 summary The proof is a gap-free stitch of four regimes. | Regime | Mathematical mechanism | Public evidence | |---|---|---| | $2\le d\le100$ | Smaller Hensel certificates over finite fields | Exact integer and finite-field verifier; primes at most 41 | | $d\equiv1\pmod3$ | Exact special point $(0,0,1)$ | Symbolic substitution and diagonal Jacobian minor | | $r\in\{0,2\}$, $33\le m<48{,}550$ | Quantitative root persistence at Gaussian-rational locators | 97,033 ordered FLINT/Arb rows, terminal receipt, no failures | | $r\in\{0,2\}$, $m\ge48{,}550$ | Uniform coefficient-defect envelope and Newton--Kantorovich theorem | Exact-rational endpoint bound $0.0029969535913992734<0.003$ and monotonicity proof | 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. ## 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 | Audience | Potential use | Required caution | |---|---|---| | Polynomial-automorphism researchers | Inspect a claimed resolution of the full $e=3$ containment column and reuse the smooth-point formulation. | The full conjecture and fixed-$e$ programme remain open. | | Algebraic geometers | Study the weighted-Jacobian factorisation and local smoothness bridge. | The Lewis--Perry--Straub translation and analytic bridge still merit independent reconstruction. | | Analysts and special-functions researchers | Examine an explicit Fourier-exponential limit with quantitative $C^1$ persistence. | The eventual constants and norm translations are producer-authored and computer assisted. | | Computer-assisted mathematics researchers | Audit a proof object that joins exact, interval and analytic regimes through one normative specification. | Replay establishes the encoded predicates and byte integrity, not an unaffiliated proof. | | Interested non-specialists | See how a theorem can combine geometry, asymptotics and exhaustive certified computation. | Candidate publication is not peer-reviewed consensus. | ## 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. Investigate the fixed-$e$ extension only after the $e=3$ result survives independent review; keep failures and changed thresholds visible. ## 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. ## 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. ## Open directions for follow-up research - Independently reconstruct the Jacobian--Euler, Fourier-limit, Newton--Kantorovich and Lewis--Perry--Straub bridges from the definitions. - Write a separate interval implementation and rerun all 97,033 immutable locators without producer intermediates. - Obtain external specialist reviews in polynomial automorphisms, algebraic geometry, asymptotic analysis and validated numerics, retaining corrections publicly. - Formalise the structural identities and persistence theorem over an explicit trusted base. - Investigate the fixed-e extension only after the e=3 proof survives independent review, publishing changed thresholds and failed routes. ## Research process and reusable methods This is prospective process metadata. It records the intended handoff and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:polydegree-full-e3-column - Attempt receipts: ep-attempt:full-e3-column-polydegree-conjecture: published / positive; measured-partial; active human minutes missing; compute minutes missing; rework minutes missing; assurance endpoint measured-partial. Work ledger: https://evidencepress.org/api/work-ledger.json - Intended aims: science - Artifact roles: research-output, method-demonstration, communication - Decision object: certificate — A gap-free all-degree proof object joining an exact smooth-point reduction, explicit Fourier-limit zeros, a rational eventual envelope and 97,033 ordered finite Arb certificates. Scope: The containment G_(d+3) contained in the closure of G_(d,4) for every integer d at least 2; not the full Polydegree Conjecture or the conjectural fixed-e extension. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Exact regime stitching (regime-stitching); Adversarial scientific controls (adversarial-controls); Explicit research-lineage reuse (research-lineage-reuse); 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 normative specification maps the exact coefficient normalisation, norms, locators, tails, Newton predicates, third-row condition, Jacobian transport and complete range order to code; the manuscript separately states the Lewis--Perry--Straub implication and analytic derivation. Remaining risks: The manuscript, specification and implementation share producer-side lineage and may share a transcription error.; The analytic tail derivation and application of the Lewis--Perry--Straub criterion require independent specialist reconstruction.; No separately authored interval checker has tested the archived locators and range stitch.. - Human judgement gates: Assess the Jacobian--Euler and Lewis--Perry--Straub bridges as mathematics rather than inferring them from successful computation.; Assess the Fourier-limit and Newton--Kantorovich derivations, constants and norm translations independently.; Judge novelty and priority against the published record beyond the dated bounded search.; Authorize scholarly attribution, public release and any later claim about the fixed-e programme or workflow impact. - Next assurance action: An unaffiliated specialist team should reconstruct the analytic and Polydegree bridges and implement the finite and eventual interval predicates independently against the immutable public inputs. - Claim ceiling: A producer-side theoretical and computer-assisted proof of the full e=3 column, with exact and interval replay and bounded public-record novelty evidence; not independent reproduction, formal verification, peer review, the full conjecture, a fixed-e theorem, absolute priority or demonstrated workflow impact. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Faster or more reliable resolution and assurance of difficult all-parameter mathematical problems in AI-assisted theoretical and computer-assisted work on the e=3 Polydegree Conjecture column; design none; comparator No prospective matched conventional research or publication workflow comparator was registered.; estimand No effect on discovery time, active human effort, compute, correction rate, proof quality or independent-assurance time was estimated.; no real-world effect evidence asserted - Parent handoffs: extends-result smooth-point-certificates-polydegree-containments; inherited claim: The derivative--Jacobian identity, integral weighted-Euler factorisation, smooth-point reformulation and smaller Hensel certificate form the structural layer used by this all-degree successor.; inherited ceiling: The parent is an anonymous unrefereed producer-side candidate with exact replay and coordinated review, not independent confirmation, external specialist review or formal verification. ## Verification status Anonymous unrefereed candidate. The main all-degree e=3 containment is presented as a theoretical and computer-assisted theorem. Exact algebra, finite-field checks, a rational eventual envelope and the complete finite Arb bridge have passed producer-side replay, and the final review requested only bounded minor revisions. No unaffiliated party has rerun the full immutable Arb package or independently reconstructed the proof; there is no separately authored checker, proof-assistant formalisation, external specialist review, editorial peer review or venue acceptance. The dated public-record search found no earlier published all-degree e=3 proof; it is not a specialist priority adjudication. ## Sources and related work - Anonymous (2026). Smooth-Point Certificates for Polydegree Containments: A Jacobian Interpretation of the Lewis--Perry--Straub Determinant. Version 0.4.1-candidate. - Lewis, D., Perry, K., & Straub, A. (2019). An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms. Journal of Pure and Applied Algebra, 223(12), 5346--5359. - Perry, K. A. (2016). Polydegree properties of polynomial automorphisms. Doctoral dissertation, The University of Alabama. - Ortega, J. M., & Rheinboldt, W. C. (1970). Iterative Solution of Nonlinear Equations in Several Variables.