bibkey: corkdisneyhogg2025jnr
authors: Josh Cork and Linden Disney-Hogg
year: 2025
title: Locations of JNR skyrmions
doi: 10.1007/s11040-025-09513-3
url: https://arxiv.org/abs/2505.00075v2
claim: Remark 13 conjectures strict positivity of the JNR triple sum (18) without the balance condition (15).
strata_touched: []
license: citation-only
triage: anchor

Locations of JNR skyrmions

Verified locator

DOI: 10.1007/s11040-025-09513-3

Source: https://arxiv.org/abs/2505.00075v2

Source statements

Appendix A, Remark 13, PDF page 20 (printed page 19):

The above proof shows that this is positive so long as Σᵢ (λᵢ²/|yᵢ|²) yᵢ = 0, however we conjecture that the sum in (18) is positive for any choice of yᵢ ∈ ℝ³∖{aᵢ}.

We have a proof of this for N = 2, for N = 3 with equal weights, and when the yᵢ are on a single sphere, but a general proof without any constraint on the yᵢ appears to be a hard combinatorial problem.

Equation (3) defines “ρ(μ) = Σ_{i=0}^N λᵢ² / |x − aᵢ + μ|²”. At μ = 0, with yᵢ = aᵢ − x, this is ρ(0) = Σᵢ λᵢ² / |yᵢ|². The quaternion norm becomes the Euclidean norm for the pure imaginary poles and point. Equation (18) is

$$S=\frac{1}{\rho (0)}\sum _{i,j,k=0}^N\frac{{\lambda }_i^2{\lambda }_j^2{\lambda }_k^2⟨y_j-y_i,y_j-y_k⟩}{(\Vert y_i{\Vert }^2+{\mu }^2)(\Vert y_j{\Vert }^2+{\mu }^2)\Vert y_j{\Vert }^2\Vert y_k{\Vert }^2}.$$

Encoding and scope

D5/S3/Analytic/JNRRemarkThirteenPositivity uses Fin (N + 1) for the poles, EuclideanSpace ℝ (Fin 3) for their carrier, positive λᵢ and μ, injective poles, and x distinct from every pole, with N ≥ 1. The source’s displayed exclusion is interpreted as x ≠ aᵢ, hence yᵢ ≠ 0, as specified in issue #19039.

The repository proof factors the two outer sums and completes a square at the β-weighted barycentre. It concerns expression (18); it asserts no equality with the source’s earlier double sum away from condition (15), and no physical realization. The source’s other theorems retain their own hypotheses.

Experiment and literature boundary

The exact and numerical comparison of (18) and the two square forms is in the-omega-institute/trureturing-experiments, docs/reports/cork-disney-hogg-2025-jnr-remark13-positivity/, commit 7259d8d6c51ea9b5c1a72bc731e94dc9681817a9. The check.py SHA-256 is 34bc783540f3264fc197e71a603340b23ad130f02f3d9595355bd26e83c299c9; python3 check.py exits 0 with RESULT PASS 0.

Issue #19039 records the literature screen, including the correction DOI 10.1007/s11040-026-09577-9. Literature completeness is ASSUMED-UNVERIFIED: Google Scholar, Semantic Scholar and native MathDB were not fully accessible in that screen. The SOS derivation was supplied by the search seat recorded there; the Lean proof does not claim independent discovery.