slug: fan-2026-circle-capacity-conjecture-5-9 bibkey: fan2026riesz doi: 10.48550/arXiv.2609.11186 url: https://arxiv.org/abs/2609.11186v1 triage: theorem motivation_gids:

• D5/S3/Geometry/Distances/FanCircleCapacity.result

Fan Conjecture 5.9: moving three-point circle capacity

Problem

Conjecture 5.9 of Qiuling Fan, Riesz capacity ratios with negative exponents, arXiv:2609.11186v1, printed page 19, states:

Suppose r ≥ 2. Let A, B, C be three points on the unit circle. Fix B and C such that they are symmetric with respect to the x-axis, having polar angles φ and −φ, respectively, where φ ∈ (π/2, 2π/3]. Move A along the circle with polar angle ψ ∈ [0, 2π − 3φ], so that BC remains the longest side of the triangle △ABC. Then the capacity of the three-point set {A, B, C} is maximized when ψ = 2π − 3φ, i.e., when AC = BC.

The formal target keeps that original domain exactly. For every real r >= 2, pi/2 < phi <= 2*pi/3, and 0 <= psi <= 2*pi-3*phi, it compares the set of actual complex unit-circle points at angles psi, phi, and -phi with the isosceles endpoint. Capacity is Fan’s negative-exponent Riesz capacity: the 1/r power of the supremum of the positive r-power distance energy over all probability measures on the actual point subtype. The target does not restrict r to integers, impose positive singleton masses, use a finite grid, or claim strictness or uniqueness.

Motivation

The frozen declaration D5/S3/Geometry/Distances/FanCircleCapacity.result is the exact formal settlement whose source identity this dossier records. The moving point is constrained by a fixed circle and a fixed longest chord; this is not the unconstrained three-point optimizer or a capacity-ratio problem. The endpoint comparison also crosses a change of optimizer support, so a proof must handle the zero-mass branch rather than silently assume three positive masses.

Gap

Fan’s Section 5.4.1 records an unsuccessful monotonicity approach, not a proof of Conjecture 5.9. Clark and Laugesen, Maximizing Riesz Capacity Ratios: Conjectures and Theorems, Theorem 8, supply the classical exact energy maximum for a three-point metric space, and their Section 7 supplies the concave-power argument used below. Those results do not state the fixed-circle, fixed-longest-chord comparison in Conjecture 5.9. The remaining gap is to bind the optimizer to all probability measures on the actual point subtype and prove that its attained value is nondecreasing along Fan’s prescribed motion, including the support transition.

Route

For a moving angle t, set x=(phi-t)/2 and y=(phi+t)/2. The source bounds give 0<x<=y<=pi-phi<=pi/2 and x+y=phi. The three chord lengths are a=2*sin(x), b=2*sin(y), and the constant c=2*sin(phi), with 0<a<=b<=c. Write A=a^r, B=b^r, C=c^r, and s=A+B-C.

First transport a probability measure on the actual three-point subtype to an ambient measure concentrated on that set, and transport it back. Finite integration then expresses its energy through the singleton masses u,v,w on the closed simplex, including zero masses, as 2*C*u*v + 2*A*v*w + 2*B*w*u. If s<=0, its maximum is C/2, attained at (1/2,1/2,0). If s>0, put D=4*A*B-s^2; the nonnegative masses

u0=A*(B+C-A)/D, v0=B*(C+A-B)/D, and w0=C*(A+B-C)/D

sum to one and attain U=2*A*B*C/D. The completed-square identity

U-energy = ((2*B*(u-u0)+s*(v-v0))^2 + D*(v-v0)^2)/(2*B)

proves the upper bound. Thus both branches are the single attained profile

U=(C/2)*(4*A*B)/(4*A*B-max(s,0)^2).

On the active branch, apply the Clark-Laugesen concave-power inequality v-u <= v^p-u^p for p>=1, 0<=u<=v<=1, and u^p+v^p>=1, with p=r/2, u=a^2/c^2, and v=b^2/c^2. Together with the chord and cotangent identities, it makes (B+C-A)*cot(y)-(A+C-B)*cot(x) nonnegative. Differentiating the unified profile gives a nonnegative derivative when s>0, zero when s<0, and zero at s=0 because max(s,0)^2 is differentiable there. Continuity and the mean-value theorem make the attained energy nondecreasing on the whole interval. The increasing 1/r power preserves the endpoint comparison, and at t=2*pi-3*phi one has b=c, exactly Fan’s isosceles endpoint.

Falsifier

A probability measure on the actual point subtype whose energy exceeds the displayed optimizer, a failure of the ambient/subtype transport, a point in the stated real parameter domain where the attained energy decreases, or a mismatch between the formal endpoint and psi=2*pi-3*phi would invalidate the route or settlement. A finite-grid computation, an integer-exponent theorem, or a proof for positive singleton masses only would not settle the registered problem. A prior proof of the same fully quantified source statement would invalidate open-problem-resolution eligibility without changing the theorem’s truth.

Evidence

The primary arXiv v1 definitions, Conjecture 5.9, and Section 5.4.1 were read at the versioned source recorded by D5/L/Geometry/fan2026riesz. The same Library note records the actual probability-measure convention, the exact parameter range, and the source’s unsuccessful approach. The preregistration is issue #9471 and quotes the source statement before the candidate proof work.

The bounded prior-work comparison covered Fan’s 2025 thesis, the relevant arXiv v1 paper sections, Clark-Laugesen Theorem 8 and Section 7, the final SIAM version of that paper, Fan’s pinned source repository, and the differently constrained triangle work identified in issue #9471. Clark-Laugesen receive credit for the three-point optimizer and concave-power step. The subtype measure equivalence, explicit attainment on the full closed simplex, unified support-transition profile, and fixed-circle derivative comparison are the repository route. The frozen result retains only propext, Classical.choice, and Quot.sound in its axiom closure.

Triage

First tier: a named conjecture in a 2026 paper, preregistered in issue #9471. admission_basis: open-problem-resolution; proof_shape: content for the single public result. Its live proof contains the full measure bridge, attained optimizer, and monotonicity argument rather than only instantiating an existing fixed-circle theorem. The three public definitions are the necessary source objects, and no bind-only companion theorem is delivered. The result is uniform in all real parameters in the source domain, not bounded enumeration, a checker, numeric reduction, or a certified finite instance; utility: none.

ASSUMED-UNVERIFIED

The prior-resolution finding is bounded to the sources and repository searches recorded in issue #9471 and the Library note. It is not an exhaustive worldwide novelty, priority, or independent-proof claim. The final metadata still requires independent review and CI; this dossier does not claim merge, issue closure, programme completion, or KPI credit. Fan’s other conjectures, capacity-ratio optimizations, strictness, uniqueness, and equality classification are outside this result.