Exact memory requirements for retaining path irreversibility in finite Markov systems

We study the additional information that must remain after labeled jumps have been read and discarded, so that the accumulated path log-likelihood ratio can be reconstructed exactly and updated under subsequent inputs. For a fixed product of homogeneous three-state rings with positive rational rates and a uniform stationary initial law, this question reduces to an integer-current coding problem.

Given the total jump count m, and no other history-dependent side information, the exact number of required output classes is C_E(m) = |E S_d(m)|, where E is the prime-exponent matrix of the rate ratios and S_d(m) is the reachable current set. If r = rank_Q E, the minimum additional fixed-length code uses r log₂(m+1) + O_E(1) bits. A recursive counter representation attains this leading term.