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

We analyse allocation, admission and post-write retention in finite-horizon linear-Gaussian noisy recurrent memories. At every horizon, the directional Fisher memory Mn satisfies operatornametrMn=N: non-normality redistributes information but cannot raise its spherical average, while normal carriers satisfy Mn=I. For bi-power-bounded carriers, we derive uniform 1/n lag bounds, identify the limit of Mn with the inverse of the classical Ces`aro 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 of magnitude in J. In a separate pre-specified study of 320 runs, trained masks followed the designated input-time objective in both carrier types, in 16 of 16 draws. These studies used development-seen carriers and are pre-specified validations, not blind holdouts. The same fixed design reproduced the objective-specific result in 16 of 16 draws on carriers unused before run commitment. Exact isolation preserved information, while a decoder fixed at its training horizon fell to chance; inverse-adjoint transport restored its sampled decisions to numerical precision.