bibkey: fiore2026squaresideddice authors: Evelyn Fiore, George D. Nasr, and Cooper Stone year: 2026 title: “Dice Relabeling using Square-Sided Dice” doi: null url: https://arxiv.org/html/2606.20311v1 claim: “Definition 6.1 defines the full cyclotomic pairs; Conjecture 6.8 explicitly includes negativity at the midpoint when p is not two.” strata_touched:

• D5/S3/Factorization/Cyclotomic/FioreNasrStoneMidpointRefutation license: CC-BY-4.0 triage: anchor

The square-sided-dice midpoint assertion

Verified locator

Evelyn Fiore, George D. Nasr and Cooper Stone, Dice Relabeling using Square-Sided Dice, arXiv:2606.20311v1, 18 June 2026, Definition 6.1, Table 2, Lemma 6.6 and Conjecture 6.8. The primary HTML at https://arxiv.org/html/2606.20311v1 has 663149 bytes and SHA-256 293426354df81fdb96ce3c34861f16a308fc299f215760752a07777f0b208ef3. The source is distributed under CC BY 4.0. The quotations and mathematical notation here are attributed to those authors; the counterexample and its interpretation below are repository derivations. No paper, experimental program or experimental data is redistributed here.

Exact assertion and domain

Section 6 treats distinct primes p, q and r and exponents a,b,c,d in {0,1,2}. Definition 6.1 gives

$$A_{abcd}(X)=X{\Phi }_p(X{)}^2{\Phi }_q(X{)}^2{\Phi }_{pq}(X{)}^a{\Phi }_{pr}(X{)}^b{\Phi }_{qr}(X{)}^c{\Phi }_{pqr}(X{)}^d,$$

$$B_{abcd}(X)=X{\Phi }_r(X{)}^2{\Phi }_{pq}(X{)}^a{\Phi }_{pr}(X{)}^b{\Phi }_{qr}(X{)}^c{\Phi }_{pqr}(X{)}^d.$$

The pair uses B with exponents 2-a,2-b,2-c,2-d. The complete Conjecture 6.8 is:

In the remaining 33 pairs of the polynomials A_abcd and B_(2-a,2-b,2-c,2-d) (including the case where p != 2 for when a=b=c=d=1), one of the two polynomials will have a negative coefficient.

The midpoint is an explicitly included clause. The pairwise distinct odd primes 3,5,11 satisfy it, including p != 2. They also satisfy p<q<r and have r maximal, so conventions about ordering cannot exclude this instance. The formal claim retains all factors and every natural coefficient index. It is the midpoint universal assertion, not a full encoding of the remaining exponent cases. Falsifying this expressly required instance falsifies the source conjecture.

Source boundaries

Table 2 duplicates (2,1,1,2); its printed count 33 does not justify an invented remaining-case predicate. Lemma 6.6’s assertion that sums of odd-prime multiples are always even fails at 11=2*3+5. The lemma is not a proof supplier. Its necessity claim for midpoint B is contradicted by the complete nonnegative factorization at 3,5,11. No corrected classification of all exponent tuples is asserted. Section 3’s coefficient convention allows zeros; nonnegative coefficients, rather than strictly positive coefficients at all degrees, are the relevant condition.

Literature boundary

The bounded source intake checked the latest arXiv version (v1), the exact title, and the predecessor arXiv:2408.10331v2 / DOI 10.54550/eca2025v5s2r9. Source-specific public Problems and issue/PR searches and private-inclusive D5/Library/Problems searches found no matching resolution in that scope. Issue #11894 concerns the different Conjecture 5.7 Q01 classification. Rate limits at OpenAlex and Semantic Scholar and unavailable search endpoints limit the search. No exhaustive priority, novelty or global absence assertion follows from these readings.