Two stability criteria for constrained phases of multi-colour Ising models: exactly six strictly closed supports at every colour count

Abstract

Version 2 withdraws a claim made in version 1. Version 1 of this note asserted that the complete four-colour Ashkin-Teller study (10.5281/zenodo.22832697) had drawn an incorrect inference about its C1 and C3 phases, and a correction to that effect was attached to that record. That assertion was wrong and is withdrawn in full. It applied one stability criterion to a result that study obtained, and explicitly states, under a different one. Nothing in that study, or in the companion colour-parity study, is in error. See WITHDRAWAL_AND_CLARIFICATION.md and section 9 of the manuscript. 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. Studies of the doubled, tripled and complete four-colour models identify such phases as strong-coupling sinks of a diamond-hierarchical renormalization group, using the criterion that the allowed mismatch shells are exactly those retaining maximal weight after one step. This note isolates a second, strictly stronger criterion - that the support be exactly reproduced - and shows that the two behave entirely differently as the colour count grows. Under support closure the inventory is finite, small, and independent of N. If the bond weight is permutation symmetric and the cell length b is odd, the support must be closed under threefold addition; translating by any of its own elements turns it into a subgroup, so it is a coset of a binary linear code. Permutation symmetry forces it to be a union of complete Hamming shells, and only six shell-unions are cosets: the zero shell, the top shell, their union, the even shells, the odd shells, and the whole space - for every N >= 3, degenerating to five at N = 2. The six are the trivial code, its nontrivial coset, the repetition code, the parity-check code and its coset, and the whole space. Under argmax closure the inventory is strictly larger and genuinely N-dependent. Enumerating it for N = 3 to 12 reproduces the census of odd-face complement-closed sinks published in the companion parity study (10.5281/zenodo.22837860) exactly at all ten colour counts, for both b = 3 and b = 5, including the irregular entry at N = 12 where {1,11} and {5,7} are sinks but {3,9} is not. The odd/even colour-count asymmetry reported in that census therefore lives entirely in the weaker criterion: the strictly closed inventory is parity-blind. The two criteria are separated by the single shells. At N = 4 the shells of mismatch 1 and 3 are argmax-closed - so the C1 and C3 phases built on them are correctly identified sinks, and their certified residual entropy interval stands - but are not support-closed, since the threefold sum 1000 + 0100 + 0010 = 1110 escapes and the support spreads to the full odd face. They are dominant-shell sinks inside the odd coset rather than additional entries in the six-element classification, which is consistent with the four-colour study's own parameterisation of the odd sector by the ratio of shell weights. Argmax closure is not merely a weakening: at N = 8 the mismatch-3 shell fails it, the argmax migrating to mismatch 1. Verification. All arithmetic is exact and rational. The Krawtchouk transform is itself put under test first, against direct enumeration of the b-fold sumset on all 486 shell sets for N = 2 to 6 and b = 3, 5 - not a formality, since the forward and inverse transforms take opposite index orders and swapping them produces plausible but wrong profiles. The coset lemma is then brute-forced over all non-empty subsets: 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 deposited script reproduces every table, asserts at each step, and runs in under a minute. Not claimed. Any error in either companion study. Any statement about which of the six is stable on a given lattice at a given dimension. Any explanation of the N-dependence of the argmax family, including the absence of {3,9} at N = 12, which is confirmed here and left open. Anything about q >= 3 states per colour, where the two lemmas do not combine in the same way. Declaration of generative AI use. This work was produced with substantial AI assistance (Claude, Anthropic). The model developed the classification, performed the verification, drafted the note, and identified the version-1 error. The author defined the research direction, made the scientific decisions, and takes full responsibility for the content, including for the withdrawn claim.