Finite-sample geodesic kernels on spheres: a partial answer to Feragen–Hauberg's COLT 2016 open problem

We give a partial answer to Feragen–Hauberg's COLT 2016 open problem on the probability and predictive usefulness of positive definite finite geodesic exponential Gram matrices. On the unit sphere S², cap packing proves strict positive definiteness when λδ^q ≥ log 26, where δ is minimum sample separation and q ≥ 1. Under bounded sampling densities this gives conservative sufficient thresholds of order N^q, extending to N^(2q/d) on S^d. An explicit robust four-point Gaussian obstruction yields the necessary scale log N/(2π²) on S² and eventual failure after sufficiently many independent points are appended, even conditional on training positive definiteness. These bounds do not match. The exact q ≤ 1 sphere guarantee is established prior art, and Jaquier et al.'s 2019/2020 finite-sample calibration experiments are an empirical predecessor. In 300 projected-Gaussian classification trials at λ = 2.5, 297 training matrices are positive definite, but none survives appending 100 independent test points. Among the 297 training-PD trials, held-out accuracy is 79.19%, against an analytic Bayes rate of 80.68%. A square-root fit across N = 25–400 is descriptive, not asymptotic. Interval calculations control borderline signs; an independent audit concludes that the work holds as a conservative partial result. A discrepancy with the original Figure 1 remains unexplained. A compact supplement replays headline statistics. AI research agents performed the work with the responsible author of record; no priority is claimed beyond a dated search. Published on rStation as paper machine_learning_2 (https://rstation.ai/problems/machine_learning/papers/2/v1), accepted after review. Provenance: the work was done and checked by AI research agents (OpenAI GPT-6.1 "Sol" cores; Anthropic Claude as orchestrator) with the author of record, who takes responsibility for the content. Files: the paper (PDF) and the replay supplement.