Criterion-dependent optimality of binary and cyclic Gray encodings under asymmetric write and retention noise

Minimizing the Hamming distance between consecutive codewords is not equivalent to maximizing every probabilistic measure of reliability. This preprint develops an exact finite-state stochastic framework for cyclic binary encodings under distinct write and retention error probabilities. It separates two reliability criteria: Historical Survival, determined by cumulative Hamming activity and Pointwise Reconstruction, determined by the complete finite-time stochastic dynamics. The study establishes seven principal scientific results: an exact stochastic model for arbitrary cyclic encodings; an exact analytical characterization of Historical Survival; a structural relation between the two reliability criteria; a complete-cycle optimality theorem for cyclic Gray encoding when write errors are more probable than retention errors; an exact Markov characterization of Pointwise Reconstruction; an exact criterion-dependent Binary–Gray ranking reversal for the two-bit counter; and exact higher-dimensional verification for B = 3 and B = 4, including the four-bit Hamiltonian-cycle and coordinate-permutation-orbit analysis. The central result is that, under the same physical error parameters and observation horizon, Gray encoding can be superior under Historical Survival while Binary encoding is superior under Pointwise Reconstruction. Thus, encoding optimality is criterion dependent rather than absolute.