A General Theory of Error Detection via Symmetry Breaking

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, both heuristically and through constructive examples, that m* depends not only on E but, critically, on the choice of σ, turning permutation selection into the strongest design lever. As a constructive proof we exhibit a decimal (n = 5) permutation σ_opt = [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 purely decimal two-check-digit construction is known to deliver this guarantee. Extended error classes (cyclic permutation, block swap) require m* > 3, confirming the E-dependence of m*. Companion Python scripts reproduce every claim by finite exhaustive verification or controlled Monte Carlo simulation; the framework is alphabet-independent and reduces check-digit design to a finite intersection problem.