Resolves a decade-old question about quantum uniformity measures: ∥·∥Q4 are norms, but the triangle inequality fails for all k≥5. Advances fault-tolerant quantum computation theory.

Resolves a decade-old question about quantum uniformity measures: ∥·∥Q4 are norms, but the triangle inequality fails for all k≥5. Advances fault-tolerant quantum computation theory.
New proof establishes the existence of ghost r-SICs—equichordal subspace configurations—by extending the twisted convolution identity to rank-r structures using finite quantum dilogarithm theory.