--- title: "Inverse-Root Support and Fourier Fusion for Polynomial Exponential Periods" date: 2026-08-20 version: "0.5.0-candidate" doi: 10.5281/zenodo.22036030 pdf: https://github.com/ipitchford/cyclicity-support-fusion-atlas/releases/download/v0.5.0-candidate/Inverse-Root-Support-and-Fourier-Fusion.pdf repository: https://github.com/ipitchford/cyclicity-support-fusion-atlas archive: https://zenodo.org/records/22036030 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Inverse-Root Support and Fourier Fusion for Polynomial Exponential Periods ## Plain-English summary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 $$ P(\xi_P(t))=t^d, \qquad \xi_P(t)=t+O(t^{-1}). $$ Split it by exponents modulo $d$: $$ \xi_P(t)=\sum_{r\in\mathbf Z/d\mathbf Z}X_r(t), \qquad \Sigma(P)=\{r:X_r\ne0\}. $$ 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 $$ \boxed{ q(P,dx)=|\Sigma(P)|=\dim_{\mathbf C}W_P^{\mathrm{red}}. } $$ 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 $$ L_c=\{\ell\in\mathbf Z/n\mathbf Z:c+m\ell\in\Sigma(P)\}. $$ If $V_a$ is the constant span of the roots in block $a$, then every block set $A$ satisfies $$ \boxed{ \dim\sum_{a\in A}V_a = \sum_{c:L_c\ne\varnothing} \operatorname{rank} \bigl(\zeta_n^{a\ell}\bigr)_{a\in A,\,\ell\in L_c}. } $$ 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 $$ P(x)=(x^4+x)^3=H(x^3), \qquad H(y)=y(y+1)^3, $$ as the obstruction to a naive additive block rule. The present candidate computes $$ \Sigma(P)=\{1,7,10\}\subset\mathbf Z/12\mathbf Z. $$ 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. | Rank | Normal form | Conditions | |---:|---|---| | 1 | $x^d$ | $d\ge2$ | | 2 | $D_d(x,a)$ | $d\ge3$, $a\ne0$ | | 2 | $x^{2m}+Ax^m$ | $m\ge2$, $A\ne0$ | | 3 | $x^{3m}+Ax^{2m}+Bx^m$ | $m\ge2$, excluding the recorded lower-rank loci | | 3 | $x^4+Ax^2+Bx$ | $B\ne0$ | | 3 | $(x^4+ax)^3$ | $a\ne0$ | 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. | Outcome | Cases | |---|---:| | Full allowed family, with matching saturating family | 45 | | Power-only certificate | 884 | | Union-of-predicted-components certificate | 223 | | Anomaly | 0 | | Missing certificate | 0 | | **Total** | **1,152** | 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: ```bash 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](https://evidencepress.org/releases/cyclicity-loci-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](https://evidencepress.org/releases/cyclicity-root-span-low-rank/) 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 | Reader | Start here | Principal caution | |---|---|---| | Twisted de Rham and irregular-period specialists | Theorems 4.3, 5.2, 6.1 and 7.3 | The proofs are unrefereed and not formally verified. | | Polynomial decomposition and monodromy researchers | Theorems 8.1-9.5 and the degree-twelve collision | Fourier compatibility must still be assessed as mathematics, not inferred from tests. | | Computer-algebra researchers | Theorem 13.1, per-case receipt and fail-closed runner | The finite atlas trusts the generator, source-to-encoding bridge, Python and SymPy. | | Future all-degree prover | Conjecture 13.2 and the quarantined descent | Do not reuse the circular single-power step; begin at the degree-thirty-two lift. | | AI research agents | `AI_INDEX.md`, `CLAIMS.json`, `CLAIM_EVIDENCE_MAP.json` and `ASSURANCE.md` | Preserve the distinction between universal proofs, finite computation, conjecture and exploratory programmes. | | Reviewers | `reviews/`, `PUBLICATION_REPAIR_NOTE.md` and `REPRODUCIBILITY.md` | Internal cross-model review and producer replay are not independent assessment. | ## 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](https://github.com/ipitchford/cyclicity-support-fusion-atlas/releases/tag/v0.5.0-candidate). The archival version is [Zenodo record 22036030](https://zenodo.org/records/22036030), DOI [10.5281/zenodo.22036030](https://doi.org/10.5281/zenodo.22036030). ## Open directions for follow-up research - Prove or refute the all-degree coefficient atlas by replacing the circular quartic and cubic lift step with a non-circular twisted-factor descent or a new monodromy argument, beginning with the degree-thirty-two lift. - Obtain focused external specialist review of the formal-moment isomorphism, fixed-sector Fourier-Gamma determinant, equality-factor argument and Ritt fibre-product classification. - Reimplement the complete degree-twenty atlas independently in another open-source computer-algebra stack and compare normalized per-case proof objects. - Formalize the support/root-span identity and finite Fourier block-rank formula in a proof assistant with a declared axiom and computer-algebra trust footprint. - Classify higher-rank Ritt diamonds and identify which abstract Fourier fusion profiles are realizable by polynomial decompositions. - Develop the generic ODE corollary into explicit minimal operators, and pursue the zero-cycle and Laurent-polynomial channel programmes without importing them into the current theorem boundary. ## 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:cyclicity-support-fusion-atlas - Attempt receipts: ep-attempt:cyclicity-support-fusion-atlas: 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, evidence-assessment, method-demonstration, communication - Decision object: reusable-method — A support-to-rank and Fourier block-fusion calculus for polynomial exponential periods, paired with a finite exact coefficient-atlas protocol. Scope: Universal support, root-span and fusion theorems for actual polynomial decompositions; finite affine coefficient classification only through degree twenty; all-degree atlas remains conjectural. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Explicit research-lineage reuse (research-lineage-reuse); Productive failure and stop receipts (productive-failure); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: discovery, assurance, publication - Semantic bridge: explicit — The normalized inverse-at-infinity coefficients are mapped to congruence support, active Kummer channels, reduced inverse-root spans, decomposition-block Fourier ranks, affine coefficient families and per-case Groebner witnesses. Remaining risks: No unaffiliated specialist has validated the formal-moment, analytic or Ritt bridges.; The coefficient-atlas source-to-encoding bridge and exact arithmetic trust Python and SymPy.; The quartic and cubic descent needed for the all-degree atlas is open.. - Human judgement gates: Assess the formal-moment and active-channel proofs as mathematics rather than inferring truth from successful replay.; Assess whether every claimed fusion profile is tied to an actual polynomial decomposition and whether equality classes really define the stated factor.; Keep the degree-twenty finite atlas distinct from Conjecture 13.2 and the first open degree-thirty-two lift.; Require a new release decision before claiming independent verification, external specialist endorsement, priority or an all-degree theorem.; Confirm anonymous scholarly attribution, component licences and public-release authority. - Next assurance action: Obtain focused unaffiliated specialist review of the formal-moment, fixed-sector and Fourier-fusion proofs, paired with an independently authored coefficient-atlas implementation and a targeted attack on the degree-thirty-two lift. - Claim ceiling: Universal written theorem candidates for inverse support, cyclic rank, root span and Fourier fusion; a producer-side finite computer-assisted atlas through degree twenty; not an all-degree atlas, independent reconstruction, formal verification, external specialist or editorial peer review, absolute priority, or demonstrated research impact. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Reusable support and fusion tools for polynomial decomposition and related period problems in Research in polynomial exponential periods, inverse-root relations, Ritt theory and computer-assisted classification; design none; comparator No matched conventional mathematical-research or publication workflow was registered.; estimand No effect on discovery time, error rate, proof quality, review effort, theorem production, reuse, uptake or citation was estimated.; no real-world effect evidence asserted - Parent handoffs: extends-result cyclicity-loci-exponential-periods; inherited claim: The predecessor introduced exact fixed-seed cyclicity loci and the stationary-phase bridge that originally connected algebraic rank to periods.; inherited ceiling: The predecessor is an anonymous unrefereed candidate with producer-side replay; it is not independent verification of this successor.; extends-result cyclicity-root-span-low-rank; inherited claim: The predecessor identified reduced inverse-root span and block fusion as the route to a decomposable low-rank classification.; inherited ceiling: The predecessor's fixed-seed identity was conditional and its decomposable classification remained open; its internal reviews and replay are not independent confirmation. ## Verification status Anonymous unrefereed theorem candidate. The universal inverse-support, cyclic-rank, reduced-root-span, fixed-sector period and Fourier block-fusion claims are presented as written theorems. The affine coefficient atlas is a finite computer-assisted theorem candidate only for degrees two through twenty. The all-degree atlas, zero-cycle programme and Laurent-polynomial programme remain respectively conjectural or exploratory. Producer-side ordinary and optimized replay passes with exact receipts and semantic failure controls. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, identified external specialist review, editorial peer review, settled novelty or priority assessment, or demonstrated field impact is claimed. ## Sources and related work - Anonymous. (2026). Fixed-Seed Cyclicity Loci for Polynomial Exponential Periods (Version 0.2.1-candidate). Zenodo. - Anonymous. (2026). Fixed-seed cyclic rank as reduced inverse-root span (Version 0.1.0-candidate). Zenodo. - Ritt, J. F. (1922). Prime and composite polynomials. Transactions of the American Mathematical Society, 23(1), 51-66. - Zieve, M. E., and Mueller, P. (2008). On Ritt's polynomial decomposition theorems. arXiv:0807.3578. - Girstmair, K. (1999). Linear relations between roots of polynomials. Acta Arithmetica, 89(1), 53-96. - Sabbah, C. (1999). On a twisted de Rham complex. Tohoku Mathematical Journal, 51(1), 125-140. - Hien, M. (2009). Periods for flat algebraic connections. Inventiones Mathematicae, 178, 1-22.