Quality Control Algorithms for Pattern Counting

What the math gives to ML

The paper develops an asymmetric certification principle: instead of exactly counting a complicated global pattern, test a small collection of block-level discrepancies that every sufficiently non-random object must violate, while random objects pass with high probability. For permutations, the certificate is a jumbledness condition on block intersection counts, backed by hypergeometric concentration and a union bound. This transfers naturally to hard mixture-of-experts routing, where token-to-expert assignments form a bipartite incidence matrix and localized concentration creates expert overload or positional routing bias. The practical adaptation is a cheap discrepancy-based router regularizer and diagnostic rather than an exact pattern-counting algorithm.