The order generated by a column of the SU(n)_k S-matrix is locally a numerical semigroup ring (Part II)

Part I determined the ring generated by a single column of the modular S-matrix of SU(n)k: at prime height p = n+k it is, locally at p, a monomial ring whose valuations form a numerical semigroup. This note asks how the columns fit together. The p-adic valuation of the discriminant of the fusion ring splits into ramification, twice the sum of the column defects, and twice a gluing term; the gluing has a closed form by orbits except for Δ(n,p) = ∑orbits δ(T), and this note gives an asymptotic expansion of that term. The splitting itself is verified computationally and not proved, and what rests on it inherits that status. The defect is a function of one thing only, of which power sums of the spectrum vanish, so the census is a question about vanishing moments. We classify the small defects: for a centred spectrum with trivial stabiliser, δ ≥ 2 is governed by two single vanishing conditions and δ ≥ 3 by exactly four pairs of them, one for each numerical semigroup of genus three. The proportion of columns with δ ≥ 2 is 2/p − 1/p2 up to an explicit error for each fixed n ≥ 5, with no factorial threshold; the mean defect per orbit is 1 + 2/p + 3/p2, the coefficient 3 being −1 from the first expansion and one unit for each of the four semigroups; and in the central range εp ≤ n ≤ (1−ε)p both hold uniformly, the first with an exponentially small error. Two classical objects turn up at the ends of that distribution. At the top, the columns of maximal defect are exactly the totally split Bring–Jerrard trinomials Xn + aX + b with ab ≠ 0, counted for n = 5 by the elliptic curve of conductor 50; the Hasse–Weil bound is quadratically miscalibrated there, and the first such column occurs at p = 67, 163 and 601 for n = 5, 6, 7, all below n!. Deep in the distribution, one of the four genus-three families is realised by the half-systems of Gauss's lemma, which give an infinite family of non-Gorenstein orders whose value semigroup ⟨3,5,7⟩ does not move with p. Separately, the centred half-systems themselves are counted by H(p) = (2(p−1)/2 + (2|p)(p−1))/p, a closed form for a sequence catalogued since 1965 and an exact relabelling of the Fermat quotient of 2.