# Verification report: OPG-37325

Title: Logarithmic Equivalence Covers of Powers of Cycles.
Author: Alper Ferudun, Mercury Software GmbH.
Date: 29 September 2026. Status: unrefereed preprint; originating-workflow
self-audit, not independent review or proof-assistant certification.

## Exact mathematical claim

For all integer k>=1 and n>=2k+2,

    ceil(log2(2k+2)) <= eq(C_n^k) <= 4 ceil(log2(k+1))+1.

If k divides n, the upper coefficient is 2 for even n/k and 3 for odd
n/k, with the same +1 term. Thus the covering number has logarithmic
order uniformly in n, contradicting the linear-growth conjecture even
in the intended arbitrarily-large-n regime. The complete-graph range is
treated separately and is not used to manufacture the counterexample.
Exact minima and optimal leading constants are not claimed.

## Analytical audit

The full derivation is retained in reproducibility/complete_proof.md.
The 15-point audit addresses threshold endpoints, clique validity,
per-layer disjointness, the unique short-arc decomposition, the single
short-block skip, the m=3 parallel-activity case, arbitrary remainders,
the rank factorization, sequence overlap, and all quantifiers.
No mathematical gap was identified by that self-audit.

## Exact implementation diagnostics

- 3,114 parameter pairs checked, including 5,523,720 host-edge incidences.
- Seven general threshold models with duplicate thresholds and both endpoints.
- Four rejection controls: added nonedge, duplicate vertex, omitted necessary
  layer, and invalid complete-graph use of the noncomplete lower witness.
- Serialized certificate (n,k)=(112,7): seven layers, 784 edges, all 6,216
  unordered vertex pairs checked by a separate certificate-consumer routine.
- Serialized certificate (1984,31): eleven layers, 61,504 edges, all
  1,967,136 unordered vertex pairs checked by that same consumer.
- For these certificates, all 136 and 2,080 lower-triangular matrix entries
  were evaluated with integer arithmetic, with nonzero diagonal of sizes
  16 and 64 respectively.
- Python 3.9.6 and Python 3.13.5 returned identical mathematical reports
  apart from the runtime field. Both reports are included.

The deterministic finite-evidence SHA-256 is
`c3f4e2b99cd5e561d9b9c180b09a6d7e4013d71871bf84a71a055946df530465`.
These diagnostics test concrete implementations; they do not prove the
infinite-parameter theorem, establish novelty or supply independent review.

## Sources, originality and typesetting

The rank argument is credited to Alon (1986). Logarithmic co-chain encodings
are credited to Golumbic, Morgenstern and Rajendraprasad (2018). The
explicit application to cycle powers was not located in the bounded search,
but a 2010 MathFest abstract treats the same topic without stating bounds.
Priority remains uncertain and no absolute-first claim is made.

The standalone source compiled in the built-in LaTeX editor and was exported
using the existing local TeX runtime. The final four-page PDF has no TeX
warnings. All four page images were visually inspected for clipping,
missing glyphs, broken references, equations and author presentation.

The public source package excludes third-party source PDFs and full texts.
PDF/source/report hashes and package members are recorded in manifest.json.
At this preparation checkpoint, the paper is NOT externally published and
has no assigned DOI. Publication must be separately confirmed from actual
public Zenodo records, file checksums, the live site and the HF comment.