Zero-Downtime Hardening of Legacy Patient Identifiers Against Transcription Errors: A Table-Free Verhoeff Formulation

Background . Patient identification in health registries relies on decimal identifiers whose integrity checks were designed for isolated systems. In a densely allocated namespace, an undetected transcription error may silently retrieve another person’s clinical record. Commutative check-digit schemes are structurally blind to certain adjacent transpositions; the Hungarian Social Security Number serves as our case study. Methods . We formalise the identified blind spot and propose a backward-compatible supplementary fold-check digit derived from a non-commutative fold over the ten-element dihedral group. We give a cycle-conjugate arithmetic realisation of the Verhoeff permutation that replaces the standard permutation and multiplication tables with two 10-element vectors, modulo-5 and modulo-10 arithmetic, and a single exclusive-or operation. To our knowledge, this is the first table-free realisation suitable for safety-critical audit. Results . Under the legacy check, 20 of the 100 ordered digit pairs are invisible to the checksum, yielding an approximately 10% upper bound on identity-altering yet formally valid errors. The proposed fold-check digit deterministically eliminates this gap for adjacent transpositions and single-digit substitutions. Monte Carlo experiments with 8,000 and 150,000 trials per configuration confirm 100% rejection of both error types. Conclusions . The hardened protocol offers minimal-footprint hardening layer: no change to the primary identifier, backward compatibility, and a deterministic guarantee against adjacent transpositions and single-digit substitutions. The cycle-conjugate formulation reduces the classical Verhoeff table footprint by roughly 90%, and the framework generalises to any legacy decimal identifier with a commutative check. The approach strengthens the identifier infrastructure on which clinical decision support and health information exchange rely, closing a foundational patient-safety gap.