Uniform stability of approximately idempotent laws on finite groups

For every $C>2$, a probability law on a finite group with sufficiently small self-convolution defect is within C times that defect of the uniform law on a subgroup, in full ¹ norm. The threshold depends only on C, independently of the group. The coefficient $2$ cannot hold at any universal positive threshold. The proof applies the corrected small-projection theorem of Zippin to obtain approximate stochastic factors, then recovers an exact permutation action and a coset partition. A local estimate improves the resulting coarse approximation to the stated linear bound. We distinguish the abelian consequence of Saeki's norm-gap theorem and record, in an appendix, an alternative linear factorization argument using Kitaev's approximate-algebra theorem, with credit to Osborne's prior formulation.