bibkey: fan2026riesz authors: Qiuling Fan year: 2026 title: Riesz capacity ratios with negative exponents doi: 10.48550/arXiv.2609.11186 url: https://arxiv.org/abs/2609.11186v1 claim: Conjecture 5.9 asks whether the three-point negative-exponent Riesz capacity is maximized at the isosceles endpoint of the stated unit-circle motion. strata_touched:

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

Riesz capacity ratios with negative exponents

Section 1.1 and equation (2) define the positive-order energy as the maximum, over all probability measures on the set, of the double integral of the positive real power of Euclidean distance. The associated negative-exponent Riesz capacity is the reciprocal-exponent power of that energy. For a finite set, equation (2) writes the same quantity as the full nonnegative mass simplex sum; zero masses are allowed.

Conjecture 5.9 on printed page 19 fixes two unit-circle points at polar angles phi and -phi, with pi / 2 < phi <= 2 pi / 3, and moves the third point through 0 <= psi <= 2 pi - 3 phi. It asserts that, for every real r >= 2, the capacity is maximized at psi = 2 pi - 3 phi, where the two longer chords are equal. Section 5.4.1 records an unsuccessful approach rather than a proof.

The formal result proves this conjecture with ProbabilityMeasure on the actual subtype of the three complex points. Every such measure is reconstructed from its three singleton masses, including zero masses, and transported to and from ambient measures concentrated on the point set. Explicit maximizing masses identify the attained energy supremum in both optimizer branches.

The three-point quadratic maximum and the real-power concavity calculation are credited to Clark and Laugesen, Maximizing Riesz Capacity Ratios: Conjectures and Theorems (SIAM Journal on Mathematical Analysis, DOI 10.1137/24M171992X), Theorem 8 and Section 7 respectively. The unified positive-part transition and the chord/cotangent derivative comparison along Fan’s fixed-circle path are the repository’s synthesis; they are not attributed to that paper. Together they show that the exact optimizer value is nondecreasing from psi to 2 pi - 3 phi, and monotonicity of the nonnegative reciprocal power yields the stated capacity inequality for every real r >= 2.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2609.11186
• URL: https://arxiv.org/abs/2609.11186v1
• Version and location: arXiv:2609.11186v1, Sections 1.1, 3, and 5.4, especially Conjecture 5.9 and Section 5.4.1.
• Retrieval: the arXiv HTML endpoint returned HTTP 200 on 22 September 2026.

Formal result

• D5/S3/Geometry/Distances/FanCircleCapacity.result proves Conjecture 5.9 with the source’s full real parameter range and all probability measures.
• The kernel statement uses the actual complex point subtype and makes no finite-grid, integer-exponent, fixed-pentagon, or positive-mass restriction.