Develop an analytic null distribution for the Intersection ECP profile

Develop an analytic null distribution for the Intersection Euler Characteristic Profile that can calibrate the profile’s hypothesis tests without permutation resampling.

Background

The paper calibrates its one-sided and paired tests using label permutations and sign flips. Although these procedures are exact, they require repeated recomputation of the profile statistic across many relabelings. The authors identify an analytic null distribution as a way to calibrate the tests without permutations, particularly in large-scale layer, epoch, and class-family sweeps.

The problem is explicitly presented as part of the paper’s remaining theoretical agenda rather than as a methodological result established by the current work.

References

sharper procedures---structured multiplicity control along the spectrum's floors, and an analytic null that would calibrate without permutations---are left to future work.

— Certified Topological Interaction in Neural Representations: Class Disentanglement Is Mostly Pairwise  (2609.08561 - Majhi, 8 Sep 2026) in Section 3, subsection “Estimation and multiplicity”; reiterated in Section 6, “Conclusion and limitations”