A General Theory of Error Detection via Symmetry Breaking

v3 update: Among major revisions, this version extends the general theory to totally anti-symmetric quasigroups, shifting the structural core from group axioms to the Latin-square property. ---------------------------------------------------------------------- We present a general framework for the design of check-digit codes over finite alphabets in which the fundamental question is not how much redundancy to add, but how to break the symmetries of the error channel. The framework rests on three elements: (1) a state group G = Dnm, where Dn is the dihedral group of order 2n matching the alphabet size q = 2n, and m is the number of check digits; (2) a phase-coupled fold F(σ,k) parameterised by an anti-symmetric permutation σ and a phase vector k; and (3) the symmetry-matching condition Stab(G, F) ∩ E = {e}, which asserts that the algebraic stabiliser of the code (its set of fold-invariant sequence transformations) has trivial intersection with the channel-realisable error family E. We prove a stabiliser monotonicity theorem: enlarging the direct product by one factor can only shrink the stabiliser, so the search over m terminates at a (possibly non-trivial) limit; whenever that limit is {e} a minimal check-digit count m* is well-defined and finitely computable. We further show, through two explicit constructions, that m* depends not only on E but, critically, on the choice of σ, turning permutation selection into the principal design parameter. As a constructive proof we exhibit a decimal (n = 5) permutation σopt = [3,8,5,2,1,4,0,6,9,7] with phase k = 4 for which m* = 2: two decimal check digits deterministically detect all four documented Damerau human-transcription error classes (substitution, adjacent transposition, twin error, jump transposition). To the author's knowledge no previously published two-check-digit construction is known to combine this four-class deterministic guarantee with full surjectivity onto the decimal check-pair space, free of excluded check values, padding conventions, and payload-length restrictions. We show that the sufficiency direction and the monotonicity theorem hold over any totally anti-symmetric quasigroup, not only the dihedral group, so the structural core rests on the Latin-square property rather than on the group axioms, and the even-alphabet restriction is removed. The framework is alphabet-independent and reduces check-digit design to a finite intersection problem.