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, 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

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

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

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

But the dimensions of the separate root blocks need not add. Put

For every cube root of unity (zeta),

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

It reduces

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. The exact archived version is Zenodo record 21864884, DOI 10.5281/zenodo.21864884.

This is a linked follow-up to Fixed-Seed Cyclicity Loci for Polynomial Exponential Periods, DOI 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.

@misc{cyclicityrootspanlowrank2026,
  title        = {Fixed-seed cyclic rank as reduced inverse-root span},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.21864884},
  url          = {https://doi.org/10.5281/zenodo.21864884},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/cyclicity-root-span-low-rank/}
}