Where the character values of Sp(2m) at a torsion element fail to generate the ring of integers

Take every irreducible character of Sp(2m) and evaluate them all at one element of finite order. They generate a ring. This note locates and describes what that ring is missing from the ring of integers — its conductor — and finds Stickelberger's ideal and a relative class number waiting there. Let ξ be a primitive q-th root of unity and let g have eigenvalues ξ±1, …, ξ±m, with 1 ≤ m < q/2. The character values generate A = Z[e1(η), …, em(η)] inside the ring of integers O = Z[ξ+ξ−1] of the real cyclotomic field, with ηj = ξj+ξ−j. Apart from known points where A = Z, it is an order, and its conductor is the subject. Four things are proved. Write h = 2m, the Coxeter number of type Cm, and q = pvq′ with p not dividing q′, d = φ(q′)/2. Which primes. Every prime of the index divides q; and for q′ outside {1,2,3,4,6}, a prime p | q divides it exactly when q′ divides h, h+1 or h+2 — the same three families of integer points at which g is conjugate to all its prime-to-q powers. The combinatorial input is a lemma on scales that also answers a question about musical tuning. What the defect looks like. At each such prime the conductor is a power of the radical, the order has a single maximal ideal above p with residue field Fp, and the first graded layer has the dimension of the cyclic module over Fp[(Z/q′)×] generated by an explicit first-moment function of the segment {±1, …, ±m}. Both lengths, a Gorenstein criterion in layers and the Cohen–Macaulay type follow. Why class numbers appear. On the stable families that first moment is Stickelberger's sawtooth, or the same sawtooth shifted by half a period — and a single identity, 2 ŝn(c) = sn(2c) − sn(c), governs both regimes. The maximal minors of the moment matrix are Sinnott's index of the Stickelberger ideal, so the conductor at p is the radical exactly when p does not divide h−(Q(ζq′)), and a simple divisor of h− gives the square of the radical and a Gorenstein local ring. On the shifted sawtooth the subgroup generated by 2 decides everything, with almost Gorenstein rings of type 2δ−1; and the index closes, vp([O:A]) = 2d−1−W1, whenever the first layer is more than half full. Below and beyond. When p divides the multiplicity there is a universal ladder of layers at the degrees P−pj, P = pvp(N), whose first term is Kontsevich's finite logarithm; and at degree P the theory descends to the first layer of a smaller order of the same family, an equality that complex conjugation — not a formal argument — is what pins down. What is not ours, said plainly. The note carries a section that states, line by line, what is proved here and what rests on the work of others: Amiot on scales; Sinnott's index formula in the form of Bernard and Kučera; Carlitz–Olson on Maillet's determinant; Kuribayashi's rank of the rectangular matrix; the Euler factor of the shifted sawtooth, which is in Kanemitsu–Kuzumaki and in Kučera; the length characterisation of Gorenstein rings after Bass, Herzog–Kunz and Huneke–Swanson; Campillo–Delgado–Kiyek on the symmetric criterion; Marseglia's formula for the type; Barucci–Fröberg on almost Gorenstein rings. Where no earlier statement was found, that is said as a search result and priority is claimed for none of it. Every number is checked, and the checks travel. Each figure and each number labelled measured or observed comes with the script that produces it and its archived output: 137 files in the ancillary archive, in a flat directory that reads and writes only inside itself, with a README mapping every number to its script and its run. Three controls run over the whole package: that nothing published quotes private correspondence, that nothing published reads a file that does not travel with it, and that every claim made by a figure caption holds against the data the figure draws. Included: the note in English (29 pp) and Spanish (30 pp), and the ancillary archive of scripts and their runs.