Amplitude modules, fused support, and a coefficient-Bautin bound for polynomial phases

Summary

This anonymous, unrefereed candidate gives three explicit results about what polynomial amplitudes and zero-cycle perturbations can do.

For nondegenerate Dickson phases, it classifies every polynomial amplitude and reduces the induced rank to a module count. Odd degree realizes exactly the even ranks; even degree realizes every rank. For the exceptional degree-twelve phase

it finds two single channels and three fused modules, proves the hidden cancellations inside those modules, and realizes every rank from zero to eleven. For a general degree-$d$ phase and zero-cycle, it proves that the ordinary coefficient Bautin ideal is generated by the first $d-1$ Melnikov orders.

These are theorem candidates with written proofs and exact internal replay. They are not independently validated theorems, and the release does not claim that three historically posed open problems have been solved.

Candidate status: Anonymous · unrefereed · internal exact replay and model-mediated editorial repair passed · no independent specialist validation, formal verification, journal peer review or settled priority.

Summary for specialists

Let $P$ be a degree-$d$ polynomial phase, let $A=G'$, and let $q(P,A\,dx)$ be the rank of the complete exponential-period system generated by $A(x)e^{sP(x)}dx$. The load-bearing bridge identifies $q$ with both the non-trace support of $G(\xi(t))$ and the reduced constant span of $G$ on a generic fibre.

For $P=D_d(x,a)$ with $a\ne0$, every $G\in\mathbb C[x]$ has a unique free $\mathbb C[P]$-module decomposition into paired Dickson summands and, when $d$ is even, one midpoint summand. If $N$ paired summands and midpoint indicator $\varepsilon$ are active, then

For the collision

the candidate displays two single composition components and three rank-three fused modules. A Newton fibre-trace lemma supplies the crucial exclusion missed by a one-way Chinese-remainder argument: the relevant generators are trace-free, so residues four and eight do not reappear. The resulting rank formula is

For a degree-$d$ phase $f$, a zero-cycle $C$ and polynomial perturbations of arbitrary fixed degree, the ordinary coefficient Bautin ideal is generated by Melnikov orders $1,\ldots,d-1$. Thus the explicitly defined coefficient index obeys $b_{\mathrm{coeff}}(m)\le m-1$. The same inequality applies to the motivating source's terse $b(m)$ only if that source uses the identical coefficient-ideal convention.

Technical mechanism: three finite-algebra reductions

The first reduction converts inverse-series residue support into finite-fibre linear algebra. Formal moments identify the twisted quotient with Kummer channels; a corrected falling-factorial Vandermonde argument computes cyclic rank; Fourier inversion then matches those channels with generic-fibre values. The manuscript adds a standard-ray Fourier--Gamma realization to connect the formal channel calculation to the complete rapid-decay period system.

The second reduction uses monodromy blocks. Central monodromy projectors descend to free $\mathbb C[P]$-modules, and every nonzero block component activates its full Fourier-character block. Dickson symmetry makes the blocks paired, with one even-degree midpoint. The exceptional Ritt phase instead has three fused three-dimensional blocks; fibre traces prove their hidden cancellations are stable under $\mathbb C[P]$-coefficients.

The third reduction controls every perturbation order at once. An all-order Lagrange--Bürmann identity expresses Melnikov coefficients in a finite quotient algebra. Differentiation is injective on the relevant zero-cycle polynomial integrals, and Cayley--Hamilton supplies a recurrence of length $d-1$. This generates the ordinary coefficient ideal from the first $d-1$ orders.

The paper also constructs a scalar operator for the complete rapid-decay period system and identifies its minimal order from inverse support. That construction is supporting infrastructure, not a fourth headline result. For a specified ray-cycle, only corrected monodromy-block bounds are retained: a Bessel example refutes the old unrestricted raw channel-count formula.

What is established in the candidate

• A written all-amplitude Dickson module decomposition and exact parity law for $a\ne0$.
• A written exceptional-Ritt module decomposition with a fibre-trace proof of the hidden cancellations.
• A written ordinary coefficient-Bautin theorem with a $d-1$ generating window for every zero-cycle and arbitrary fixed perturbation degree.
• Exact support-rank, Dickson, exceptional-module, coefficient-Bautin and operator receipts over rational inputs.
• A complete package replay with 13 command executions and 14 byte-identical artifact comparisons.
• Five deliberate mutation controls and matching ordinary/optimized 9-test JUnit inventories.
• A five-role internal model-mediated substantive review, consolidated repair, one bounded confirmation and deterministic closeout.

Evidence and assurance

Availability and producer-side internal replay pass. The public GitHub tag and Zenodo record expose the scientific ZIP, PDF, accessible MathML HTML, source manuscript, manifest, release binding, JUnit parity and closeout receipt under persistent identities.

Data and environment reproducibility are partial. Exact commands, tested-host versions and receipts are public, but the environment is non-hermetic and no unaffiliated cross-platform rebuild has been reported. Semantic validation and novelty assessment are also partial: the release contains explicit claims, proof locations, a source-convention audit and a bounded primary-source search, but the universal arguments still require independent mathematical reconstruction and the search cannot settle priority.

Independent rerun, independent reimplementation, proof-assistant formalization, external specialist review and editorial peer review are not assessed. The five internal roles were Editor-in-Chief, methodology, domain, applications and Devil's Advocate. Their reports are internal model-mediated quality control, not independent review.

What the release does not establish

• It does not establish that three historically posed open problems have been solved.
• It does not establish independent reconstruction, specialist acceptance, journal peer review or formal verification.
• It does not establish absolute novelty or priority.
• It does not unconditionally identify the manuscript's $b_{\mathrm{coeff}}(m)$ with the motivating source's terse $b(m)$.
• It does not give a real planar limit-cycle bound.
• It does not cover arbitrary contour systems or restore the refuted equality between a chosen cycle's raw channel count and scalar differential order.
• It does not treat exact replay, internal review, public hashes, a DOI or publication as theorem certification.

Who should care, and why

Why this result matters

The Dickson theorem replaces amplitude-by-amplitude computation with a parity rule. Once the free module components of $G$ are known, the rank follows immediately. The exceptional collision shows what changes when decomposition symmetries fuse channels: the answer is still finite and explicit, but only after a trace argument detects cancellation invisible to block membership alone.

The Bautin theorem supplies a different kind of compression. An infinite perturbation series has an ordinary coefficient ideal generated by a fixed initial window determined only by the phase degree. That gives a concrete stopping rule for the stated coefficient convention and exposes the exact semantic question that remains before transferring the result to the motivating notation.

Together, the three results make the Atlas support calculus operational beyond the original fixed-seed setting. They turn inverse support into amplitude modules, attainable ranks and a finite ideal-generation bound while keeping the analytic, source-convention and assurance boundaries visible.

How to inspect the release

Begin with the 23-page paper or the accessible HTML companion. Then inspect CLAIMS.json, the release binding, and the exact scientific-package/ subtree in the tagged repository.

The DOI record preserves nine release files. Complete Atlas-dependent replay additionally requires the unchanged Atlas v0.7 archive identified in REPLAY_INPUTS.json; it is linked to its existing release and is not redistributed in this record.

Run the package checker in ordinary and optimized Python, then verify evidence/junit/junit-parity.json. This tests package semantics and the retained finite predicates. An independent mathematical review should reconstruct the generic-specialization, standard-ray, monodromy-projector, fibre-trace and Bautin-recurrence arguments from the definitions before relying on the implementation.

The most valuable next projects

1. Reconstruct the support-rank theorem and standard-ray realization independently from the public statement.
2. Reprove the Dickson invariant-field decomposition and exceptional fibre-trace cancellation without using the producer's reducers.
3. Reimplement the support, fused-module and coefficient-Bautin checks in a separately authored CAS stack.
4. Formalize the generic-specialization, monodromy-projector and Cayley--Hamilton bridges in a proof assistant.
5. Obtain an authoritative specification of the motivating source's $b(m)$ convention and either prove or reject identity with $b_{\mathrm{coeff}}(m)$.
6. Derive the exact differential-module contribution of each monodromy block for controlled cycle systems.

What is in the public package

The versioned scientific ZIP contains the manuscript, PDF and generated TeX; theorem dossiers; exact Python programs and JSON receipts; source, convention and novelty audits; model-mediated editorial reports and repair matrix; environment and replay records; licences; a 72-entry scientific manifest; and the stored exact-root integrity receipt.

The release also supplies standalone accessible HTML, ordinary and optimized JUnit, a parity receipt, deterministic preflight and closeout evidence, a whole-unit manifest and an immutable release binding. Evidence Press art, Open Graph media, transcript, synthetic-voice audio and thumbnail are communication aids. They do not add mathematical evidence.

@misc{amplitudemodulesfusedsupportbautinbound2026,
  title        = {Amplitude modules, fused support, and a coefficient-Bautin bound for polynomial phases},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22143919},
  url          = {https://doi.org/10.5281/zenodo.22143919},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/amplitude-modules-fused-support-bautin-bound/}
}