Stable twisted equilibria on random geometric tori in every fixed dimension

Unrefereed research candidate. In one sentence: identical Kuramoto oscillators scattered at random on a flat torus of any fixed dimension have stable twisted equilibria near every integer winding, which proves the twisted-equilibrium conjecture of Cirelli, Groisman, Huang and Vivas for every kernel in their class with |z|K(z) bounded. Place n identical oscillators at independent uniform points of the flat torus Td and couple points closer than εn with weight K(z/εn). If εn → 0 and nεnd+2/log n → ∞, then with high probability, for every integer twist k with |k| ≤ c/(εn+δn), where δn = (log n/(nεnd+2))1/2, there is an asymptotically stable equilibrium whose phase differs from k·x by at most C|k|δn at every vertex (Theorem 2.1). The kernel need only be even, supported in the unit ball, positive almost everywhere near the origin and have sup|z|K(z) finite: this covers the hard-radius graph, every bounded kernel and |z|−α, 0 < α ≤ 1. All twists k in Zd are covered, not only coordinate twists. Further results: at least c(εn+δn)−d distinct stable constructed states; a basin of fixed size; for bounded radial kernels, a heat-equation limit uniform over all times; and stable weighted harmonic lifts under smooth nonuniform sampling (bounded kernels). The key tool is a sup-norm bound for the inverse graph Laplacian uniform over all conductances above a bounded backbone. The archive contains the manuscript (PDF and LaTeX source), an exact standard-library checker of classical cycle twisted-state spectra, the numerical illustrations with their script and records, verification and claim records, preserved development drafts, two unrelated supplementary research notes kept for provenance, a rating-free internal review summary, CI and SHA-256 sums. Assurance: written informal proofs; internal AI-agent review and proof audits only (not peer review, independent reproduction or formal verification); kernels with |z|K unbounded and the connectivity regime nεd ~ log n remain open; constants are not explicit; the numerics are small illustrations outside the theorem's regime. Creator: Anonymous. Text and figures CC0-1.0; code MIT.