slug: fiore-nasr-stone-2026-midpoint-negativity bibkey: fiore2026squaresideddice doi: null url: https://arxiv.org/html/2606.20311v1 triage: theorem motivation_gids:

• D5/S3/Factorization/Cyclotomic/FioreNasrStoneMidpointRefutation.result

An odd midpoint refutes Fiore-Nasr-Stone Conjecture 6.8

Problem

The complete Conjecture 6.8 of fiore2026squaresideddice states:

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.

Definition 6.1 uses distinct natural primes p,q,r and exponents a,b,c,d in {0,1,2}. At the midpoint a=b=c=d=1 both complementary tuples are (1,1,1,1). The two complete integer polynomials are

$$A(p,q,r)=X{\Phi }_p^2{\Phi }_q^2{\Phi }_{pq}{\Phi }_{pr}{\Phi }_{qr}{\Phi }_{pqr},$$

$$B(p,q,r)=X{\Phi }_r^2{\Phi }_{pq}{\Phi }_{pr}{\Phi }_{qr}{\Phi }_{pqr}.$$

The literal closed claim : Prop universally quantifies over natural p,q,r with Nat.Prime p, Nat.Prime q, Nat.Prime r, p != q, p != r, q != r and p != 2, and asserts that some natural-index coefficient of A or of B is strictly negative. There is no ordering premise, prefix bound or scalar proxy.

Motivation

The source expressly places this midpoint in Conjecture 6.8. At (3,5,11;1,1,1,1), all domain hypotheses hold, with all primes odd and r maximal. Thus the whole source conjecture implies negativity for this specific pair. The formal midpoint universal assertion also implies that same instance. Nonnegativity of both complete polynomials therefore refutes both assertions without an equivalence between their domains.

Gap

Table 2 repeats (2,1,1,2); its literal distinct rows do not support the printed remaining-case count 33. Lemma 6.6 claims that odd-prime multiples always have even sums, but 11=2*3+5. Neither this invalid argument nor a silently corrected remaining-case domain supplies the refutation. The expressly included midpoint clause fixes the target independently of those defects.

Route

For the mathematical abbreviation $G_m(X)={\sum }_{0\le j<m}{X}^j$, cyclotomic expansion gives

$$A(3,5,11)=X{G}_3(X)G_5(X)G_3(X^{11})G_5(X^{33}),$$

$$B(3,5,11)=X\left(\sum _{s\in S}{X}^s\right)G_{11}(X^{15}),S=\{0,3,5,6,8,9,10,12,13,15,18\}.$$

The sum over S is exactly ${\Phi }_{11}{\Phi }_{15}$. All these sums have nonnegative coefficients. Coefficient convolution preserves this property; expansion X to X^k inserts zeros. The identities prove the condition at all natural indices, including the zero tails, inside the sole result.

Falsifier

A failure of either complete factorization, a negative coefficient at any index, or a source convention excluding (3,5,11;1,1,1,1) defeats this route. The source explicitly includes that instance, and no change to its prime domain or factors is permitted.

Evidence

The formal endpoint is D5/S3/Factorization/Cyclotomic/FioreNasrStoneMidpointRefutation.result, with literal type Not claim. The only other declarations are midpointA, midpointB and the closed Prop claim. Intermediate identities remain inside result. Direct suppliers are the frozen GeometricSpectrumFactorization.geometric_sum_eq_cyclotomic_product, Mathlib’s Polynomial.cyclotomic_expand_eq_cyclotomic_mul, Counterexample.cyclotomic_15 and standard expansion/coefficient laws. The source’s Lemma 6.6 is not used.

The prospective issue is 15471. The source HTML is pinned by the SHA-256 in the Library note. The source and literature intake contains no matching valid resolution in its bounded search; that reading does not establish global absence or priority.

Triage

Tier 1: a recent named conjecture. The endpoint uses the external named open-problem-resolution basis and has proof_shape: bind-only: existing suppliers, logical composition and normalization give the counterinstance. No escape witness or new general positivity result is claimed. Certified-instance utility is the actual refutes edge from result to claim. The Scribe attaches the Refuted claim to this sole result.

What the settlement shows

Proved within result: the common implementation (3,5,11) satisfies every prime-domain hypothesis and both complete midpoint polynomials have nonnegative coefficients at every natural index. The mechanism is cancellation in ${\Phi }_{11}{\Phi }_{15}$ followed by nonnegative geometric factors, so parity alone cannot force a negative coefficient.

Open: a corrected necessary-and-sufficient condition for midpoint B and the classification of the other exponent tuples. The positive-factor mechanism remains valid for this instance and does not contradict Corollary 6.3, whose stated coefficient test does not exclude it. No additional case or repaired theorem is claimed here.

The necessity direction of Lemma 6.6 and the midpoint exclusion in Conjecture 6.8 fail at this instance. Results about other exponent tuples and earlier prime-power cases are not changed or classified by result; any use of that midpoint exclusion must be reassessed against the counterexample.

ASSUMED-UNVERIFIED

The complete remaining-case assertion is not encoded. Literature coverage is bounded, with OpenAlex/Semantic Scholar rate limits and unavailable endpoints. Physical dice realization, exhaustive priority and a classification of other cases are outside the formal statement.