bibkey: misawa2025spherical authors: Ryutaro Misawa, Yusaku Nishimura year: 2025 title: Spherical Designs on S¹ of Finite Harmonic Strength doi: 10.48550/arXiv.2505.06893 url: https://arxiv.org/abs/2505.06893v2 claim: Conjecture 3.3 asserts N({p,q},2) = 5 for all distinct integers p,q greater than one. strata_touched:

• D5/S3/Geometry/MisawaHarmonicStrengthPairRefutation license: citation-only triage: anchor

Spherical Designs on S¹ of Finite Harmonic Strength

Conjecture 3.3 on printed page 6 states verbatim:

Conjecture 3.3. Let p ≠ q be integers with p, q > 1. Then N({p, q}, 2) = 5.

The same page defines N(T, 2) := min{|X| | X ⊂ S¹, Hst(X) = T} for a nonempty finite set T ⊂ ℕ. Printed pages 2–3 identify S¹ with {z ∈ ℂ | |z| = 1}, define P_k(X) := Σ_{x∈X} x^k, and give the working form Hst(X) = {k ∈ ℕ | P_k(X) = 0}. Lemma 3.6 is on printed pages 7–8; printed page 11 restates the determination of N({p,q},2) as Problem 3.15.

For five unit-modulus points with P_2 = P_4 = 0, square the points to obtain five values y_i. Their first two power sums vanish. Conjugation and y_i conj(y_i) = 1 then give the polynomial identity forcing the third power sum to vanish, hence P_6 = 0. Therefore a five-point set cannot have harmonic strength exactly {2,4}. This refutes the universal conjecture without determining N({2,4},2).

The symbolic polynomial certificate has Groebner remainder zero. Independent numerical readings found all 400 of 400 five-point BFGS runs converging to P_2 = P_4 = 0 with |P_6| <= 2.1e-12; six-point controls reached values up to approximately 6. These are readings, not the proof.

The arXiv record retains Conjecture 3.3 in version 2 dated 30 June 2026. The same authors’ arXiv:2607.01761 does not settle the finite two-element case. Semantic Scholar reported zero citations. Google Scholar and MathSciNet were not verified.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2505.06893
• URL: https://arxiv.org/abs/2505.06893v2
• Version and location: arXiv:2505.06893v2, printed pages 2–8 and 11.