Six constrained supports: a complete classification for permutation-symmetric multi-colour Ising models

Multi-colour Ashkin-Teller models place N Ising colours on each site and support ordered phases defined by a constraint on the nearest-neighbour mismatch number. Recent work identified several such phases in the doubled, tripled and complete four-colour models, raising the question of whether higher colour counts yield a richer supply. They do not. If the bond weight is permutation symmetric in the colours and the renormalization cell length is odd, the set D of bond-difference vectors carrying weight at a strong-coupling sink must satisfy D + D + D contained in D. Translating by any element of D produces a set containing 0 and closed under addition, so D is a coset of a binary linear code. Permutation symmetry then forces D to be a union of complete Hamming shells, and only six shell-unions are cosets. The constrained supports are therefore exactly {0}, the top shell, {0} together with the top shell, the even shells, the odd shells, and the whole space - for every N >= 3, with no dependence on N. The six are recognisable as the trivial code, its nontrivial coset, the repetition code, the parity-check code and its coset, and the whole space. At N = 2 the list degenerates to five. Verification. The coset lemma was checked by brute force over all non-empty subsets of (Z_2)^N: at N = 4 that is 65,535 subsets, of which 307 are closed under threefold addition and every one is a coset, with zero counterexamples. The shell enumeration was run for N = 2 to 12 and returns five supports at N = 2 and exactly six at every N from 3 to 12. Both checks are deterministic and reproduce in under a minute from the deposited script. Consequence, and a correction. The complete four-colour study (10.5281/zenodo.22832697) reports two phases, C1 and C3, as non-coset shell constraints supported on the single shells of mismatch 1 and 3. Those shells are indeed non-cosets, but they are not closed under the recursion: one threefold sum already escapes, since 1000 + 0100 + 0010 = 1110 has mismatch 3, and the closure of either single shell is the full odd coset. C1 and C3 are therefore not additional constrained supports but limit points inside the odd face, distinguished by which shell carries the dominant weight. This is consistent with that study's own parameterisation of the odd sector by the ratio of shell weights. The phases and their certified entropy are not in question; only the claim that their supports are a distinct non-coset kind of constraint. Scope. Odd cell length, permutation-symmetric weights, and Ising colours only. The classification does not say which of the six is stable on a given lattice or dimension, which remains the dynamical question addressed in the companion studies. For q >= 3 states per colour a Hamming shell is no longer a coset and the argument does not carry over; that case is open. Companion records: 10.5281/zenodo.22832697 and 10.5281/zenodo.22837860. Preprint. Not peer reviewed. This work was produced with substantial AI assistance (Claude, Anthropic). The author takes full responsibility for the content.