Beyond Mod-97: Phase-Coupled D5×D5 Folds for Two-Digit Decimal Checksums with Exact Bounds

Abstract Verhoeff’s classical decimal check-digit scheme embeds a check alphabet into the dihedral group D 5 and detects every adjacent transposition of distinct digits through an antisymmetry condition on the permutation family. Single-digit constructions provide roughly log 2 10 ≈ 3.32 bits of redundancy, which is insufficient when identifiers are densely allocated, while the classical alternative ISO 7064 mod-11 requires an eleventh symbol and is therefore unsuitable for purely decimal identifier spaces. We construct a two-digit decimal checksum on the direct product G = D 5 × D5; its 100 elements correspond bijectively to two decimal check digits without any non-decimal residue. A phase parameter k̸≡ 0 (mod 10) couples two copies of the Verhoeff permutation family into position-dependent contributions Ψ (k) i (d) ∈ G, and the resulting fold detects every adjacent transposition of distinct digits and every single-digit substitution. Adjacent-transposition detection follows algebraically from antisymmetry on the first factor of G, confirmed exhaustively over all 9 000 relevant instances. Surjectivity onto G holds for sequence lengths n ≥ 3 and every non-degenerate k, and exact preimage counts show that the undetected uniform-noise probability approaches 10 −2 as the sequence length grows. We are not aware of a previously published two-digit decimal checksum that simultaneously provides adjacent-transposition detection, singledigit substitution detection, decimal purity, and a formally proven random-noise miss bound. All proofs are either closed-form algebraic arguments or finite exhaustive verifications reproduced by a companion script.