Obtain uniform control of sample-artifact scales

Establish uniform control of the measure of the sample-artifact scale set in the population-profile recovery theorem as the sample size increases.

Background

The recovery guarantee for the Intersection ECP excludes an exceptional set of scales consisting of neighborhoods of population critical values and sample-artifact scales. The cited foundations result does not provide uniform control of the measure of the latter set as sample size grows.

Such control is needed to strengthen the interpretation of the sampled profile as a population-level object over a range of scales rather than only away from an uncontrolled exceptional subset.

References

Uniform control of $|E_M|$ in the sample size is left open in the foundations paper (their Remark~5.9); Theorem~\ref{thm:false} below carries it as a hypothesis.

— Certified Topological Interaction in Neural Representations: Class Disentanglement Is Mostly Pairwise  (2609.08561 - Majhi, 8 Sep 2026) in Appendix A, subsection “Estimation”