Inverse-Root Support and Fourier Fusion for Polynomial Exponential Periods

Plain-English summary

A polynomial exponential integral can generate a differential equation when it is differentiated repeatedly. This release asks how much of that differential system is generated by one fixed starting form, $dx$.

The answer is encoded by the inverse of the polynomial near infinity. Split that inverse series into congruence channels. The candidate proves that the number of nonzero channels is exactly the fixed-seed cyclic rank and exactly the number of independent inverse-root functions after their common average is removed. Unlike the preceding release, this equality no longer depends on an imported stationary-phase bridge.

The same channels solve the compositional problem. For every actual factorisation $P=R\circ Q$, a finite Fourier-matrix formula computes the dimension of every sum of root-block spans. It detects when blocks are independent, partly fused, or identical; identical blocks expose a genuine coarser polynomial factor. The rule is compatible with every complete decomposition and every Ritt move between decompositions.

A separate exact computation translates the support classification into a coefficient atlas through degree twenty. It checks 1,152 cases in degrees five through twenty, after direct arguments handle degrees two through four. The all-degree atlas remains a conjecture: the first uncovered quartic lift occurs in degree thirty-two.

Status: anonymous, unrefereed theorem candidate. The universal support and Fourier-fusion statements are written proofs; the coefficient atlas is a finite computer-assisted theorem candidate only through degree twenty. No unaffiliated reproduction, proof-assistant formalisation, external specialist review, journal peer review, or novelty or priority determination is claimed.

The central invariant

After affine normalisation, let the inverse-at-infinity series be determined by

Split it by exponents modulo $d$:

If $q(P,dx)$ is the dimension generated by repeatedly applying the $s$-connection to $[dx]$, and $W_P^{\mathrm{red}}$ is the reduced constant span of the inverse roots of $P(X)-z$, the principal identity is

The proof constructs a formal moment isomorphism from the twisted de Rham module to rank-one Kummer channels at $s=0$. The fixed seed is nonzero in exactly the channels indexed by $\Sigma(P)$, and a leading-term Wronskian proves their cyclic independence. This algebraic route removes stationary phase from the rank theorem.

The paper then supplies the analytic realisation separately. On every sufficiently narrow $s$-sector, fixed rapid-decay contours give a period matrix whose leading term is an explicit Fourier matrix multiplied by nonzero Gamma factors. Its determinant is nonzero, so differentiation, integration by parts, and separation of the de Rham classes do not rely on moving cycles.

The Fourier law for block fusion

Suppose $P=R\circ Q$, with $m=\deg Q$, $n=\deg R$, and $d=mn$. The roots of $P(X)-z$ split into $n$ blocks coming from the inverse branches of $R$. For $c\in\mathbf Z/m\mathbf Z$, define

If $V_a$ is the constant span of the roots in block $a$, then every block set $A$ satisfies

This is more informative than a total-rank formula. It gives every pairwise intersection and every partial block sum. It also yields two structural consequences:

1. if block spans are equal, their equivalence classes are monodromy-invariant and define a canonical coarser polynomial right factor; and
2. along a complete decomposition, projected support ranks form a monotone chain, while a Ritt move changes one intermediate projection subject to an exact fibre-product constraint.

At global rank at most three, these restrictions leave exactly eleven unordered Ritt rank diamonds.

The degree-twelve collision, completely resolved

The predecessor release isolated

as the obstruction to a naive additive block rule. The present candidate computes

Projection modulo four gives profile $[1,1,1]$: the three quartic block spans are equal. Projection modulo three gives profile $[3]$: after the Ritt move, the four cubic-block lines span the same three-dimensional space. The apparent exception is therefore the generating low-rank Ritt collision, not an unclassified anomaly.

What the coefficient atlas says

For $2\le d\le20$, fixed-seed rank at most three occurs exactly in the following affine families, after their stated lower-rank loci are removed.

The cubic-over-power row has rank one at $A=B=0$. Its only rank-two locus is $m=2$, $B=A^2/4$, $A\ne0$. For quartics, $B=0$, $A\ne0$ has rank two and the origin has rank one. The list classifies affine families; it does not assert unique parameter coordinates.

For each degree from five through twenty, the computation enumerates every support set of size at most three that contains the normalised class $1$. It constructs exact confinement equations over $\mathbf Q$, builds a Gröbner basis through order $4d$, checks stability through order $6d$, and records a radical-containment witness.

The load-bearing elimination and reductions use exact rational arithmetic. Sampling against predicted families uses a fixed 61-bit prime as an additional containment control; it is not substituted for the rational certificates.

The open all-degree problem

The same family list in arbitrary degree is Conjecture 13.2, not a theorem of this release. A proposed quartic lift assumes that a product of twisted quartic factors collapses to one $m$-th power, then uses that collapse to prove the divisibility that would justify it. The analogous cubic step has the same defect.

Exact elimination closes the quartic lift for $2\le m\le7$. The first uncovered case is $m=8$, of total degree thirty-two. A field-changing next result would therefore be a non-circular twisted-factor descent, or a new monodromy argument that closes the degree-thirty-two lift and then scales.

What is classical and what is claimed here

The candidate uses classical twisted de Rham theory, irregular periods, linear relations among polynomial roots, discrete Fourier analysis, and Ritt's polynomial-decomposition theory. Those foundations are not claimed as new.

The claimed contribution is their fixed-seed synthesis:

• the direct formal-moment proof of the support/cyclic-rank/root-span identity;
• the fixed-sector Fourier-Gamma determinant that removes the moving-cycle gap;
• the all-block Fourier rank formula and canonical equality factor;
• the complete low-rank Ritt table and resolution of the degree-twelve collision; and
• the bounded affine coefficient atlas with per-case exact witnesses.

The package's literature audit is targeted rather than exhaustive. Failure to locate the same synthesis is not a proof of novelty or priority.

Evidence and replay boundary

The release gate runs from the package root:

python3 run_core_checks.py

It verifies the complete manifest, runs the structural, formal-fraction, period, exact-rank, support/rank, fusion, heredity and semantic-negative-control tracks under ordinary and optimised Python, regenerates all 1,152 atlas cases, and compares the outputs with frozen receipts. An anomaly, a missing certificate, a receipt mismatch, a manifest mismatch, or an unlisted file is a hard failure.

The publication candidate also includes:

• an exact Fraction audit that imports neither the production module nor a computer-algebra library;
• 278 support/rank comparisons, 200 heredity pairs and structured fusion cases;
• injected anomaly, missing-certificate, support, Fourier, receipt and manifest corruptions that must be detected;
• a post-review portability note explaining why the semantic receipt records the Python 3.14 series rather than a volatile patch number; and
• a deterministic archive builder and complete SHA-256 manifest.

These are strong producer-side fault controls. They do not turn successful execution into a proof that the mathematical encoding is correct.

What is not established

• No unaffiliated researcher has rerun the immutable package or independently reconstructed the implementation.
• No proof assistant has checked the formal-moment, fixed-sector, fusion, Ritt, or atlas arguments.
• The retained adversarial review and confirmation re-review are internal and model-assisted, not external specialist sign-off.
• No journal or comparable venue has conducted editorial peer review.
• The environment is pinned at the package level but has no container, Nix, or Guix lock.
• Novelty, priority, downstream adoption, and research-workflow impact have not been established.
• The generic period-ODE statement has a non-annihilation boundary. The zero-cycle and Laurent-polynomial sections are research directions, not closed applications.

Relationship to the two predecessor releases

This release is an additive successor to two immutable candidates:

1. Fixed-Seed Cyclicity Loci for Polynomial Exponential Periods introduced exact fixed-seed rank loci and the analytic bridge used at that stage.
2. Fixed-seed cyclic rank as reduced inverse-root span identified inverse-root span and block fusion, completed the indecomposable low-rank picture under the inherited bridge, and isolated the decomposable obstruction.

The present paper removes that analytic conditionality from the algebraic rank identity and classifies fusion for every actual decomposition and Ritt move. It does not rewrite or retroactively upgrade either predecessor's assurance.

Who should read what

The most valuable next projects

1. Close or refute the degree-thirty-two quartic lift with a non-circular factor argument, then determine whether the descent extends to all degrees.
2. Commission an identified specialist review focused on the formal-moment isomorphism, the fixed-sector determinant, the equality-factor step and the Ritt fibre-product constraint.
3. Reimplement the complete atlas in another open-source CAS without importing the production reducers, and compare normalized proof objects case by case.
4. Formalize the support/root-span identity and finite Fourier block-rank formula in a proof assistant with an explicit trust report.
5. Classify higher-rank Ritt diamonds and decide which abstract fusion profiles are polynomially realizable.
6. Convert the generic ODE-order corollary into explicit minimal operators; develop zero-cycle and Laurent variants as separate, newly gated projects.

What is in the package

• A 32-page tagged PDF, canonical Markdown, DOCX and HTML.
• Machine-readable claims, evidence mappings, assurance, status, provenance, citation and release metadata.
• Exact support, Fourier, fusion, heredity, Ritt and coefficient-atlas code.
• Frozen exact-rank and 1,152-case receipts, with deterministic replay.
• The complete round-one review, response, revision tracking and confirmation re-review.
• A chronological audit of all supplied current and historical materials.
• Preserved predecessor drafts, failed all-degree descent and exploratory zero-cycle and Laurent programmes under source-history/.
• Complete SHA-256 manifests, licensing records and deterministic archive tools.

The scholarly creator is Anonymous. Ian Pitchford is the repository maintainer and publisher, not the scholarly author. Original prose and data are dedicated under CC0 1.0; original code is MIT-licensed; third-party works retain their own rights.

The immutable candidate is available from the GitHub release. The archival version is Zenodo record 22036030, DOI 10.5281/zenodo.22036030.

@misc{cyclicitysupportfusionatlas2026,
  title        = {Inverse-Root Support and Fourier Fusion for Polynomial Exponential Periods},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22036030},
  url          = {https://doi.org/10.5281/zenodo.22036030},
  version      = {0.5.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/cyclicity-support-fusion-atlas/}
}