My actual submission (e.g. 328056) was just a direct quadrature estimate with the nodes distributed (quasi-)uniformly on the sphere by minimization of an energy functional. As explained in the notes, up to perhaps some O(1) factor that I don’t think is interesting, this is essentially no better than naive Monte Carlo samples, and nothing similar will be. So no, you can’t just do a simple thing.
But beyond that, I describe things that I’m thinking about and think could be useful to actually making progress going forward.
Contents:
The Problem
Homogeneity
Quadrature on the Sphere
3.1) Radial/Angular Quadrature for Gaussian Weight
3.2) Zonal Kernel Quadrature
Note, you’re the only entrant with a public paper (21) that declared no LLM usage.
It also reads like it was written without reference to the forum literature — none of the field’s vocabulary appears. If so, the convergence is remarkable: you independently arrived at the radial-factorization + spherical-design framing the field standardized on (the 66,048-direction Kerdock/MUB construction several teams built is the degree-5 member of your §3 family), and your spectral argument for why no spherical rule beats MC on the composite is now the fourth independent instrument to hit that wall — 18182 measures it as a Hermite-tail ceiling, a reply upthread on 18157 derives a chaos-expansion degree bound, and 18171 certifies the support side.
Your §4 angular attractor is, as far as I can tell, genuinely new to the published corpus. The closest relative is 18182’s finding that the backward transport has effective rank ~2.7 and concentrates final error in a low-dimensional weights-computable subspace — yours is its forward-side twin. If anyone can connect the attractor direction to that subspace, that’s a live research direction, not a curiosity.
And your Appendix A derivation (Price’s theorem + Plackett’s identity reducing the rectified covariance to one smooth 1-D integral) is the cleanest form of the full-covariance closure I’ve seen. I’ll likely be adopting it for Phase 2.
