Inverse-Root Support, Polynomial Amplitudes, and Laurent Moment Channels

Summary: From inverse support to moment channels

Version 0.7 presents the programme as one 25-page argument. It retains the all-degree fixed-seed classification from version 0.6, then replaces the seed $dx$ by every polynomial amplitude $A(x)\,dx$, classifies vanishing on a specified zero-cycle, specialises the polynomial moment theorem to every Atlas phase with $q(P)\leq3$, and gives an exact local-channel criterion for Laurent phases.

The result is a public unrefereed theorem candidate. The fixed-seed and endpoint classifications remain relative to named imported theorems. The Laurent result is exact for each specified input, but it is not a finite classification of every Laurent pair.

What the combined release adds

This is not merely the short extension placed beside the earlier paper. The primary PDF contains the complete fixed-seed argument and the new amplitude, zero-cycle, endpoint and Laurent sections under one numbering scheme, reference list and limitation statement.

The fixed-seed backbone

For a monic centred polynomial $P$ of degree $d$, let

and split the inverse series into congruence channels

If $\Sigma(P)$ is the set of nonzero channels, the fixed-seed theorem states

The last term is the reduced constant span of the generic inverse roots. For a decomposition $P=R\circ Q$ with $m=\deg Q$, residue projection gives

Equality is the full-fusion condition. Relative to the named primitive-monodromy and finite-projective-group classifications, following that condition through every decomposition and Ritt move leaves six affine families at rank at most three: powers, Dickson polynomials, quadratic- and cubic-over-power phases, indecomposable quartics, and the degree-twelve quartic collision. Proper rank-preserving outer extensions reduce to power towers, aligned Dickson towers, one aligned square, and one cubic extension of the exceptional quartic.

The version 0.6 repair remains load-bearing. A draft uniform Kummer-kernel claim failed on an even-exponent repeated-root stratum. The released proof preserves the counterexample, uses a coefficient-stratified theorem off that stratum, and handles the exceptional aligned square separately. Arbitrary mixed Dickson chains are treated by a direct invariant-field argument rather than extrapolation from a bounded grid.

The polynomial-amplitude support theorem

Now take any polynomial amplitude $A(x)\,dx$ and choose $G$ with $G'=A$. Expand the primitive on the same inverse branch:

Let $\Sigma(P;G)$ contain the nonzero residue classes $r\in\{1,\ldots,d-1\}$ occurring among the exponents with $c_n\neq0$. Changing the additive constant in $G$ affects only the omitted trace channel.

The combined paper proves

This is more than a change of notation. A phase can have low fixed-seed rank while another amplitude generates more channels. For example, $P=(x^4+x)^3$ has $q(P,dx)=3$, but $G=x^4$ gives $\Sigma(P;G)=\{1,4,7,10\}$ and amplitude rank four.

The rank-zero class is especially simple:

Equivalently, $A(x)=P'(x)H'(P(x))$ for some polynomial $H$. This is a universal criterion for polynomial phases and amplitudes; it does not require $q(P,dx)\leq3$.

Exact vanishing on a specified zero-cycle

Let $n=(n_0,\ldots,n_{d-1})$ satisfy $\sum_jn_j=0$, and define its discrete Fourier transform by

Fourier inversion and the disjoint Laurent supports of the active channels give the exact test

This answers a specified phase–amplitude–cycle problem without moment sampling. It does not assert that the broader balanced-cycle literature is composition-only.

Persistent endpoint moments for every Atlas phase

For endpoints $a,b$ and $G'=A$, consider

The peer-reviewed theorem of Pakovich and Muzychuk says that all $M_k$ vanish precisely when $G$ is a sum of compositions $H_j\circ W_j$ over right factors $P=R_j\circ W_j$ satisfying $W_j(a)=W_j(b)$. Version 0.7 combines that theorem with the Atlas right-factor list.

When $a\neq b$ and $P(a)=P(b)$, the solution spaces are:

If $P(a)\neq P(b)$, only the zero amplitude survives. Lower-rank overlaps use the earliest applicable Atlas row. The table is exact in the combined framework, but its assurance cannot exceed either imported ingredient.

Laurent phases: an exact local channel

Let

and normalise $c_{-m}=1$. There is a unique formal local inverse $\phi(\tau)=\tau(1+O(\tau))$ satisfying

For every Laurent-polynomial one-form $A(z)\,dz$ and every $k\geq1$, formal residue invariance gives

Thus persistent Laurent moment vanishing is exactly the vanishing of this infinite coefficient subsequence. The formula includes logarithmic amplitudes, exact forms and deck-symmetry cancellations.

Why there is no finite all-Laurent atlas

The local criterion is exact, but it is not automatically a finite structural classification. An indecomposable example of Pakovich, Pech and Zvonkin has a persistent-solution space that is a free module of rank five over $\mathbf C[L]$. Composition alone therefore cannot describe every Laurent solution.

The release does not:

• infer all-$k$ vanishing from a finite zero prefix;
• promote inherited small-bidegree tables to a theorem;
• claim that every persistent Laurent solution is a composition solution; or
• claim a finite all-Laurent normal-form atlas.

Instead, it supplies the exact coefficient object against which future finite classifications must be proved.

What the checks establish

The immutable v0.7 package supplies two replay stacks. The fixed-seed stack includes sixteen scoped Lean theorems, ten exact global CAS checks, ordinary and optimised parity, inherited bounded-atlas checks, and adversarial mutation controls. The amplitude/Laurent stack compares Krylov rank with inverse-support extraction, direct Laurent multiplication with local-inverse coefficients, and specified zero-cycle and endpoint examples. Ordinary and optimised runs produce byte-identical receipts.

The reader-facing release adds:

• a complete 338-file manifest and deterministic 339-member archive;
• a 25-page PDF with 36 bookmarks and a content-derived identifier;
• static KaTeX rendering, qpdf validation and fail-closed raw-TeX preflight;
• page-by-page visual QA, including every equation and table hotspot;
• clean-extraction replay;
• a successful public Linux workflow; and
• seven byte-identical assets on GitHub and Zenodo.

These checks establish strong producer-side consistency, replay and availability. They do not establish that every proof bridge or imported classification is correct.

Assurance boundary

• No unaffiliated researcher has reported rerunning or reimplementing v0.7.
• Lean checks selected fixed-seed consequences, not the complete fixed-seed, amplitude, endpoint or Laurent proofs.
• The monodromy, finite-projective-group and composition-kernel inputs are imported; the Pakovich–Muzychuk endpoint theorem is also imported.
• Internal model-assisted re-audits and producer-side CAS diversity are not external specialist review.
• The literature audit is broad and source-verified, but not exhaustive in subscription MathSciNet, every language, unpublished work or universal citation-forward closure.
• No journal peer review, absolute priority, adoption or field-impact claim is made.

The calibrated decision remains: GO for public specialist circulation as an unrefereed theorem candidate; HOLD for promotion as an externally established theorem.

Relationship to the immutable predecessors

This is an additive successor. It does not delete or rewrite:

1. Fixed-Seed Cyclicity Loci for Polynomial Exponential Periods;
2. Fixed-seed cyclic rank as reduced inverse-root span;
3. version 0.5, whose coefficient atlas ended at degree twenty;
4. version 0.6, which supplied the all-degree fixed-seed theorem candidate; or
5. the preserved short version 0.7 amplitude/Laurent companion.

The combined manuscript is the primary v0.7 reading path because it contains the entire argument in one document. Historical releases, receipts and PDFs remain separately identifiable.

Where to inspect the result

The most valuable next step is an unaffiliated specialist audit spanning both the imported fixed-seed interfaces and the new amplitude, endpoint and Laurent arguments, followed by an independently authored implementation and deeper proof-assistant formalisation.

The scholarly creator is Anonymous. Operational maintenance and publication roles are recorded in the machine-readable provenance. Original research prose and data are dedicated under CC0 1.0; original code and tests are MIT-licensed; third-party works retain their upstream terms.

The exact candidate is available from the GitHub prerelease. The archival version is Zenodo record 22049564, DOI 10.5281/zenodo.22049564.

@misc{cyclicitysupportfusionatlas2026,
  title        = {Inverse-Root Support, Polynomial Amplitudes, and Laurent Moment Channels},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22049564},
  url          = {https://doi.org/10.5281/zenodo.22049564},
  version      = {0.7.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/cyclicity-support-fusion-atlas/}
}