Finite-Horizon Fisher Memory in Two-Sided Power-Bounded Recurrent Systems

Jeonghoon Lee

We analyse allocation, admission and post-write retention in finite-horizon linear-Gaussian noisy recurrent memories. At every horizon, the directional Fisher memory $M_n$ satisfies $\operatorname{tr}M_n=N$: non-normality redistributes information but cannot raise its spherical average, while normal carriers satisfy $M_n=I$. For bi-power-bounded carriers, we derive uniform $1/n$ lag bounds, identify the limit of $M_n$ with the inverse of the classical Cesàro asymptotic limit of $W^\top$, and give finite-horizon error bounds. A time-varying coupling defines an end-to-end store operator. The writer-optimal direction need not be store-optimal. After writing ends, an invertible hold preserves the full stored Fisher matrix. Additive contamination bounded by $α$ times the closure covariance retains at least $1/(1+α)$ of that matrix; a covariance-aware decoder attains the corresponding accuracy. With recurrent carriers held fixed, training input masks and linear readouts approached the task-specific optimum in 160 runs, with median normalized Rayleigh efficiency above $0.998$. Binary accuracy matched the Gaussian prediction to mean absolute error below $0.002$ over more than four orders…