slug: dalfo-fiol-reyes-three-quarter-conjecture-24 bibkey: dalfofiolreyes2026threequarters doi: 10.61091/um128-09 url: https://arxiv.org/html/2609.33718v1 triage: theorem motivation_gids:

• D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture
• D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture24Refutation

Dalfó-Fiol-Reyes Conjecture 2.4

Problem

Conjecture 2.4 of Dalfó, Fiol, and Reyes, A note on three-quarters circulant digraphs, section 2, quantifies over every integer k >= 1. It specifies N=k^2+4k-2, steps a=1 and b=-k-4, and lattice columns (k+2,1-k) and (2,k). It says these data give a tile of a three-quarters digraph of sector diameter k. The paper defines the sector distance using exactly the three congruences m*a+n*b, -m*a+n*b, and m*a-n*b for m,n >= 0.

Motivation

The conjecture proposes one lattice and two-step construction for all k >= 1. The paper defines a digraph’s arcs as a set of ordered pairs; its CD construction uses the generating set {a,b} and calls the graph regular of degree two in section 1 and property P1 of section 2. Under that convention, the quantified graph assertion requires two distinct outgoing neighbors.

Gap

At k=1, the specified data give N=3, a=1, and b=-5=1 in ZMod 3. The set {z+a,z+b} has cardinality one for every vertex z, contradicting degree two. The sector construction at this boundary still has exact radius one, and the two displayed lattice vectors still generate the congruence kernel. The paper’s separate TQ(5,1,2) example for k=1 does not satisfy Conjecture 2.4’s specified N,a,b.

Route

ThreeQuarterCirculantConjecture defines SectorReach with the source’s three sign sectors. Its construction results are kernel_eq_generated, sector_cover, sector_radius_lower, and generator_regular_for_k_ge_two. The latter also gives the cardinality-two out-neighbor set for k >= 2.

For every k >= 2, the construction has the stated lattice kernel and determinant, every residue is reached by radius k, residue k is absent at radius k-1, and the two step residues are distinct and nonzero. Thus the k >= 2 corrected version has the intended degree-two generator condition; the sector and lattice conclusions also hold at k=1.

Falsifier

The k=1 one-neighbor graph falsifies the original k >= 1 degree-two assertion under the paper’s set-of-arcs convention. This refutation would not apply to a reading that counts coincident generators as two parallel arcs; that reading requires a different graph convention. The counterexample does not falsify the sector-distance or lattice formulas.

Evidence

ThreeQuarterCirculantConjecture24Refutation.claim states the quantified lattice, sector, and degree conditions; its result proves ¬ claim from the k=1 one-neighbor contradiction. The five named public theorems have scoped Lean builds and accepted axiom-closure checks recorded in #11275. The module is frozen, Scribe emission succeeds, and local push preflight is green; required remote CI is evaluated on the delivery PR.

Triage

This is an open-problem-resolution of the preregistered first-tier claim in #11225. The source-bound escape registration remains open in audit #11275. The graph conclusion depends on the source’s set-of-arcs and degree-two convention; the k >= 2 construction satisfies that condition.

ASSUMED-UNVERIFIED

The preregistration’s arXiv exact-phrase, Crossref title, Google Formal Conjectures code, and repository issue searches were bounded; OpenAlex was unavailable because its unauthenticated quota was exhausted. They do not establish exhaustive publication priority. No author clarification was checked on whether the intended graph convention permits coincident steps at k=1; the refutation is conditional on the paper’s stated set-of-arcs and degree-two reading, not on an unverified Lean calculation.