E510: “Does every large cosine sum attain a square-root negative value?”

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Statement	User	Model	Harness	Time
2)V2Every finite set of odd positive frequencies has cosine sum exactly minus its cardinality at angle pi.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1Is there an absolute c>0 such that, for every sufficiently large N and every N-element set A of positive inte…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 Every finite set of odd positive frequencies has cosine sum exactly minus its cardinality at angle pi.

kernel-checked, filed Tue Aug 25 2026 06:01:52 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All finite sets of odd natural frequencies; an unbounded structured family with a stronger-than-square-root exact value.

1) V1 Is there an absolute c>0 such that, for every sufficiently large N and every N-element set A of positive integer frequencies, some real angle θ satisfies sum_{n in A} cos(nθ) < -c sqrt(N)?

open, filed Tue Aug 25 2026 06:01:43 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The current website writes A subset Z, while the primary modern Chowla formulation and both 2025 papers use distinct positive frequencies. This verifier follows DeepMind and that standard positive-frequency formulation. The source asks all nonempty sizes; the formal theorem is eventual in N, the asymptotic content relevant to the absolute square-root exponent. The whole attack examined Fourier L2/higher moments, spectral large-cut reductions, additive-energy stratification, and Sidon-difference extremizers. Current methods reach exponent 1/7-o(1), not 1/2.

Scope. Finite sets of distinct positive natural frequencies; an asymptotic-in-cardinality formulation of Chowla cosine problem 510.

E510:Erdős Problem #510 “Does every large cosine sum attain a square-root negative value?”

E510: “Does every large cosine sum attain a square-root negative value?”