Problem, prior work, and significance assessment
Henry Zweiman. September 9, 2026. Internal assessment of one proposed research manuscript, not an external priority certificate.
Published problem and solved scope
The Amit--Ashurst conjecture asks whether every nonempty fibre of every k-variable word on a finite nilpotent group G has at least |G|^(k-1) tuples. It is Ashurst's 2012 thesis Conjecture 6.2.1, restated in Camina--Cocke--Thillaisundaram (2023), Section 1, and Chlebus--Cocke--Ho (2024), Question 4.2. The manuscript proves this assertion for every metacyclic p-group, hence every nilpotent metacyclic group, and for every elementary-abelian-by-cyclic p-group. These are broad cases of an established open conjecture, not a new conjecture invented solely for this manuscript.
The common mechanism is a theorem for scalar principal-unit actions over finite commutative chain rings. A separate transfer within the same proof removes splitting when the quotient is cyclic. The general nilpotent-group conjecture is not solved. No statement about nonnilpotent metacyclic groups is made.
Primary-source comparisons
| Source and inspected version | Scope verified | Relation to this manuscript |
|---|---|---|
| Camina--Cocke--Thillaisundaram, published 2023 article, Theorem A, Sections 2--4 | All words for p-groups with a cyclic maximal subgroup; preliminary arguments in a broader metacyclic setup | Closest predecessor. Fixed-coordinate linearization and polynomial methods are credited. Our theorem removes the cyclic-maximal restriction. |
| Levy, arXiv:1101.4286v1, complete Proposition 1.3 and proof | Identity-fibre transfer for split extensions with abelian kernel | Supplies a classical conditioning argument; does not state the needed bound for every attained value. |
| Aichinger--Moosbauer, arXiv:1912.08448v1, Definition 2.1 and Lemmas 2.2, 3.1 | Functional degree through differences/augmentation ideals | The framework is classical and is explicitly credited. No originality is claimed for finite-difference calculus. |
| de las Heras--Toti--Vannacci, arXiv:2607.11605v1, introduction, Theorem A, Proposition B, Corollaries C--E | Selected Engel words in cyclic wreath products, finer estimates, and profinite applications | The present all-word result is broader in the word variable but does not replace their finer estimates or solve their profinite-section question. |
| Chlebus--Cocke--Ho, published 2024 article, Section 4.1 | Restates the conjecture as Question 4.2; records cyclic-maximal and certain class-two cases; asks a separate recursive-arity Question 4.3 | Supports the distinction between finite computations and all-word proofs. No all-metacyclic theorem is asserted in this inspected section. |
| Kishnani--Kulshrestha, arXiv:2309.09010v1, previously inspected preprint and current abstract | Bounds and word equivalences in nilpotency class two | The current result permits arbitrary class. The corrected 2024 final body remains unread, so no claim of a complete final-version comparison is made. |
| Blanchi, June 2026 preliminary note, Sections 2--3, 6, 9 and scope | Relative marked word observables, cyclic modular actions, linear lifting and ranks | Overlap in linearization is acknowledged; the inspected page does not state the all-word probability theorem. The manuscript's linearization already has earlier primary antecedents. |
The 2023 final article's title alone could suggest a complete metacyclic result; its actual main theorem has the cyclic-maximal hypothesis. The example C16 semidirect C4 with action 3 has class four, exponent sixteen and no cyclic subgroup of order 32. It is therefore outside that main theorem and outside class-two results. The manuscript proves these structural facts directly.
Priority search and its limits
The focused search included the exact conjecture name with metacyclic, abelian-by-cyclic, elementary abelian, chain ring, and 2024--2026; author publication pages; the 2023 paper's indexed citing works; and the July 2026 wreath-product paper. One citation-index query found only Kishnani--Kulshrestha; the index is incomplete, as the separately located Chlebus--Cocke--Ho article illustrates. No source inspected in this audit states the two main theorems in their full scope.
The 2024 Kent State seminar title about latest metacyclic results was located, but its listing gives no theorem statement. It does not establish either a priority conflict or novelty. The final corrected Kishnani--Kulshrestha body was not retrieved in the earlier investigation; available scope is class two. These limits are recorded rather than silently treated as exhaustive clearance. No author contact or private communication was attempted.
Why this is a substantial single contribution
The all-metacyclic result removes a real subgroup-index restriction from a published theorem. The elementary-abelian-by-cyclic result covers arbitrary module rank and groups of unbounded class, including every cyclic regular wreath product. The chain-ring estimate retains the image size of the exponent-sum map, enabling the nonsplit transfer. Its exact description of the word image over each quotient value may support later work on word distributions.
These related statements constitute one paper. A citation count cannot be predicted or guaranteed. Significance is assessed from the established conjecture, removal of hypotheses, and reusable method, not from the number of computations or pages.
Admission decision
Admit as one complete, internally checked proposed research preprint after artifact QA. The proof is self-contained and no unresolved mathematical step has been identified in the present review. Public posting is authorized by the research goal. Historical priority, expert correctness review, and journal readiness are not independently certified; the public collection must retain its AI-assistance and review-status disclosure. This decision increments only the distinct-public-manuscript count after a verified push. It does not certify the total number of significant solved open problems or complete the 45-paper goal.