--- title: "Fixed-seed cyclic rank as reduced inverse-root span" date: 2026-08-09 version: "0.1.0-candidate" doi: 10.5281/zenodo.21864884 pdf: https://github.com/ipitchford/cyclicity-root-span-low-rank/releases/download/v0.1.0-candidate/paper.pdf repository: https://github.com/ipitchford/cyclicity-root-span-low-rank archive: https://zenodo.org/records/21864884 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Fixed-seed cyclic rank as reduced inverse-root span ## Summary An exponential integral can generate a differential equation when it is differentiated repeatedly. This release asks a deliberately narrow question: **how many genuinely different directions are produced by one chosen starting form?** The candidate's answer is simple to state. Look at the inverse branches of the polynomial phase, remove the average shared by all of them, and count the independent functions that remain. Conditional on the stationary-phase bridge established in the [parent Evidence Press release](https://evidencepress.org/releases/cyclicity-loci-exponential-periods/), that count is exactly the fixed-seed cyclic rank. For polynomial phases that cannot be decomposed into lower-degree maps, this gives a complete low-rank classification in every degree: - powers have rank one; - Chebyshev phases have rank two; and - indecomposable quartics have rank three. The package also includes a finite exact calculator for trusted rational polynomial inputs. The result does **not** complete the decomposable case. A degree-twelve example has rank three because symmetry makes three normally separate root blocks coincide. The remaining problem is to understand exactly when this kind of block fusion occurs under polynomial composition. > **Status:** anonymous, unrefereed candidate. Producer-side replay passes. > Independent reproduction, formal verification, external specialist review, > conventional peer review, and novelty or priority determination have not > occurred. ## The idea without the machinery Suppose a polynomial equation (P(x)=z) has inverse branches (x_1(z),ldots,x_d(z)). Translating the source variable removes their common average, so that $$ x_1(z)+cdots+x_d(z)=0. $$ Some of the remaining inverse-root functions may still satisfy constant linear relations. Let (W_P^{\mathrm{red}}) be their span after the common trace has been removed. The central conditional identity is $$ q(P,dx)=\dim_{\mathbf C}W_P^{\mathrm{red}}. $$ Here (q(P,dx)) is the dimension generated by repeatedly differentiating the fixed class ([dx]) in the polynomial's twisted de Rham system. The new conductor-to-dimension step uses classical representation-theoretic machinery; it is not presented as a new general representation theorem. The candidate's claimed contribution is the fixed-seed application, its low-rank consequences, the exact calculator, and the compositional obstruction. ## What the three indecomposable cases mean | Rank | Phase, up to affine changes | Plain-English reason | |---|---|---| | 1 | (x^d) | All inverse branches differ only by roots of unity, so they span one direction. | | 2 | (T_d(x)), the Chebyshev polynomial | The inverse branches collapse to a two-dimensional root span. | | 3 | Any indecomposable quartic | The quartic's reduced inverse-root span has its full dimension of three. | Every other indecomposable polynomial phase has rank at least four, conditional on the imported bridge. ## Why composition is harder Right-factor heredity gives a useful necessary condition: if (P=R\circ Q), then $$ \dim W_Q^{\mathrm{red}}\leq \dim W_P^{\mathrm{red}}. $$ But the dimensions of the separate root blocks need not add. Put $$ Q(x)=x^4+x, \qquad P(x)=Q(x)^3=x^3(x^3+1)^3. $$ For every cube root of unity (zeta), $$ Q(\zeta x)=\zeta Q(x). $$ That symmetry turns the three outer blocks into scalar copies of one another. The degree-twelve phase therefore has rank three rather than the larger rank a naive additive atlas would predict. This exact counterexample is why the page does not advertise a complete arbitrary-polynomial classification. ## What the calculator does For trusted (P,A\in\mathbf Q[x]), the supplied program works in $$ \mathbf Q(s)[x],dx \big/ \{(f'+sP'f),dx:f\in\mathbf Q(s)[x]\}. $$ It reduces $$ [A,dx], \nabla_s[A,dx], \nabla_s^2[A,dx], \ldots $$ to a finite exact Krylov matrix. It returns a nonzero minor when the iterates are independent and the first exact relation when dependence appears. This is an implementation result for rational, trusted input; it is not a security claim or a substitute for the structural proof. ## What was checked - The public package has a complete 86-entry static manifest, including nested audit manifests. - Ordinary Python and `python -O` produce byte-identical top-level replay receipts. - Exact Krylov, composition-mutation, fixed-target and separately written clean-room control tracks pass. - The ten-page PDF rebuilds byte-for-byte and was checked for syntax, embedded text, layout and visible clipping. - GitHub and Zenodo publish the same ZIP, paper, manifest and receipt bytes. - Internal fixed-text mathematical reviews and scoped prior-art searches are included in the archive. These checks establish availability, package integrity and producer-side replay. They do not establish independent correctness, novelty or priority. ## What was not checked - No unaffiliated researcher has rerun or independently reimplemented the package. - No proof assistant has checked the stationary-phase bridge or the low-rank theorem. - No signed external specialist report or journal peer review exists. - The scoped searches did not establish novelty or priority. - The complete decomposable rank-at-most-three atlas remains open. - The PDF is searchable and visually inspected, but it is not a tagged PDF. ## Contributions and disclosure The scholarly creator is cited as **Anonymous**. AI systems contributed to mathematical development, exact computation, literature work, drafting, adversarial review, revision and packaging. Ian Pitchford supplied research direction, mediated review and correction, authorized publication, and acts as repository maintainer and publisher. Internal AI review and system agreement are not independent verification. The audio briefing uses an AI-generated voice. It repeats the page's summary and is not additional mathematical evidence. ## The most valuable next work 1. Obtain signed external specialist reviews of the fixed-seed stationary-phase bridge and the root-relation argument. 2. Reproduce the archived package and reimplement the calculator independently in another open-source computer-algebra stack. 3. Classify equivariant block fusion across complete polynomial decompositions and Ritt moves. 4. Formalize the imported bridge, conductor-span identity and indecomposable classification in a proof assistant. 5. Extend the criterion from (dx) to nonconstant amplitudes, relative or logarithmic families, and multivariate twisted complexes. ## Paper, archive and parent release The immutable candidate package, PDF and replay receipts are available from [GitHub](https://github.com/ipitchford/cyclicity-root-span-low-rank/releases/tag/v0.1.0-candidate). The exact archived version is [Zenodo record 21864884](https://zenodo.org/records/21864884), DOI [10.5281/zenodo.21864884](https://doi.org/10.5281/zenodo.21864884). This is a linked follow-up to [Fixed-Seed Cyclicity Loci for Polynomial Exponential Periods](https://evidencepress.org/releases/cyclicity-loci-exponential-periods/), DOI [10.5281/zenodo.21853682](https://doi.org/10.5281/zenodo.21853682). Original release prose and public records are dedicated to the public domain under CC0 1.0. Original executable code is released under the MIT License. Third-party works retain their original rights. ## Open directions for follow-up research - Obtain signed external specialist reviews of the fixed-seed stationary-phase bridge and the root-relation and polynomial-monodromy arguments. - Reproduce the exact package and reimplement the rank calculator independently in another open-source computer-algebra stack. - Classify equivariant root-block fusion under complete polynomial decompositions and Ritt moves, including monomial collisions such as (x^4+x)^3. - Formalize the stationary-phase dependency, conductor-span equality and indecomposable low-rank classification in a proof assistant with a reported axiom footprint. - Extend the fixed-seed criterion beyond dx to nonconstant amplitudes, relative or logarithmic families, and multivariate twisted complexes. ## Verification status Anonymous, unrefereed candidate, explicitly conditional on the fixed-seed stationary-phase bridge inherited from the parent release. The versioned paper and evidence package are public under DOI 10.5281/zenodo.21864884. Producer-side ordinary and optimized replay receipts are byte-identical, and the archived bytes match the GitHub release. Internal AI reviews found no material contradiction in the stated scope. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, signed external specialist review, conventional peer review, or novelty or priority determination is claimed. The indecomposable q<=3 classification is complete under the bridge; the decomposable classification remains open. ## Sources and related work - Anonymous. (2026). Fixed-Seed Cyclicity Loci for Polynomial Exponential Periods (Version 0.2.1-candidate). Zenodo. - Sabbah, C. (1999). On a twisted de Rham complex. Tohoku Mathematical Journal, 51, 125-140. - Muller, P. (2002). Finiteness results for Hilbert's irreducibility theorem. Annales de l'Institut Fourier, 52, 983-1015. - Fried, M. D. (1999). Variables separated polynomials, the genus 0 problem and moduli spaces. In Number Theory in Progress, 169-228. - Alvarez, M., Bravo, J. L., and Mardesic, P. (2013). Vanishing Abelian integrals on zero-dimensional cycles. Proceedings of the London Mathematical Society, 107, 1302-1330.