Ambient Gaussian Landmarks and a Unit-Circle Obstruction to Nyström Approximation

Alper Ferudun, Mercury Software GmbH. English unrefereed preprint, version 1.0, 24 October 2026.

For k(x,y)=1+x^T y and m ambient standard Gaussian landmarks in R^n, the manuscript derives an exact chi-square intercept law and a finite-sample lower bound on diagonal Nyström error. When m=o(n), the unit-query error tends to 2, even for data on a compact algebraic unit circle. More generally, if m/n tends to alpha, the approximant converges uniformly on that circle to alpha k. Three suitable on-circle landmarks instead reconstruct the rank-three restricted kernel exactly.

This gives a full counterexample to the displayed global-constant tail bound on page 2501 of Oberwolfach Report 44/2014. It is a complete scoped obstruction result, not a claim to have resolved every suitably modified polynomial-kernel guarantee or the report's separate Gaussian-kernel problem. Classical feature-span and rotational-projection ingredients are credited. No absolute novelty or priority certification, formal verification or independent peer review is claimed.

Publication is paused by the author. This package has no assigned DOI and has not been uploaded or announced.
