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The Boundary of Conjecture 2.4

Abstract

Conjecture 2.4’s literal positive-integer quantifier conflicts with its degree-two graph convention at k equal to one.

Definition 1.1 (The literal source claim).

Lean statement: D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture24Refutation.claim

Formalization. D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture24Refutation.claim (✓ std3).

Citation. C. Dalfo and M. A. Fiol and M. A. Reyes (2026). A note on three-quarters circulant digraphs. DOI: 10.61091/um128-09. URL: https://arxiv.org/html/2609.33718v1.

Commentary.

For every k at least one, the proposed order, two lattice columns, full radius-k coverage, exact-radius witness, and two-element outgoing neighbor set all hold simultaneously. The final clause is the source’s degree-two digraph requirement with arcs treated as a set.

Theorem 1.2 (The one-neighbor boundary retains its sector and lattice properties).

Lean statement: D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture24Refutation.boundary_facts

Proof. Machine-checked in Lean as D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture24Refutation.boundary_facts (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. C. Dalfo and M. A. Fiol and M. A. Reyes (2026). A note on three-quarters circulant digraphs. DOI: 10.61091/um128-09. URL: https://arxiv.org/html/2609.33718v1.

Commentary.

At k=1 the order is three, both steps are residue one, and each vertex has one distinct outgoing neighbor. Radius-one sector coverage, the exact-radius witness, and the lattice kernel identity still hold.

Theorem 1.3 (The degree-two assertion fails at k equal to one).

Lean statement: D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture24Refutation.result

Proof. Machine-checked in Lean as D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture24Refutation.result (✓ std3). ∎

Resolves. Problems/dalfo-fiol-reyes-three-quarter-conjecture-24 (refuted) by D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture24Refutation.result.

Source. Repository-derived.

Acknowledgement. C. Dalfo and M. A. Fiol and M. A. Reyes (2026). A note on three-quarters circulant digraphs. DOI: 10.61091/um128-09. URL: https://arxiv.org/html/2609.33718v1.

Commentary.

At k=1 the order is three and both proposed steps are residue one. Every vertex therefore has one distinct outgoing neighbor. This refutes the literal degree-two graph assertion; it does not refute the sector-distance or lattice clauses, which hold at k=1.

References