Exact certificates for the noncommutative-Minkowski unitary-categorification criterion: two fusion rings, two degenerate corners, and the exclusion list recomputed

Abstract

Lim (arXiv:2601.13490) derives from a noncommutative Minkowski integral inequality a necessary condition for a fusion ring to be the Grothendieck ring of a unitary fusion category: for every triple of basis elements, every exponent 1 ≤ p<∞ and every nonnegative vector a, an explicit inequality between two expressions in the structure constants and the quantum dimensions must hold. He searched for violations by floating-point gradient descent over the 28,450 fusion rings of the Vercleyen–Slingerland database, excluded 5,793 of them in 97 days of machine time, published no list, and reports that part of the data was lost. We observe that for a positive integer p and a ∈ {0,1}ˢ the two sides of the inequality are algebraic in the dimensions with no root left to take on the left, so that a violation can be decided exactly. For the two rings the source prints — the rank-7 integral ring it names as the simplest of the 23 exclusions that the other criteria it tests do not reach, and the rank-4 ring with dimensions (1,tfrac(1+√(13))2,1,1) — we give exact certificates at p=2, valid for every positive character of the ring: the two sides are 2 and sqrt2 for the first, and √(4-δ) and √(δ-1) with δ²=δ+3 for the second, where the violation is exactly δ<5/2. We also record that the criterion is an identity at p=1 and at a ≡ 1, two corners of the source's sampling window where its objective is identically 1, and two controls. These statements are verified in Lean 4 against Mathlib. Outside the formal development, an exact deterministic sweep of the current database (25,331 rings) certifies 6,205 exclusions with p ≤ 10; on the 11,739 rings of ranks 2,3,4,6,7,8,9, where the database and the source's table agree cell for cell, it certifies 3,556 against the source's 3,119. The list is a lower bound on what the criterion excludes, not a classification.

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