Auditing Finite-Lag Statistics of Recurrent Representations

Hidden-state trajectories are computational traces that record how recurrent representations evolve as networks process sequences. When exact state repeats are unavailable, one way to estimate conditional motion is to borrow successors from nearby states, yielding a finite-bandwidth successor smoother. However, this procedure can change the displacement being measured. At a median-distance bandwidth, write-phase displacement energy is about $1.88\times$ the exact value for GRU and LSTM. Elman instead shows near agreement in total energy despite large, opposing finite-bandwidth contributions. We therefore compute the conditional-mean energy fraction exactly on repeat-copy. Training reverses the architecture ordering of this fraction, with GRU and LSTM rising to about $94\%$ while Elman falls to about $55\%$. The centered remainder is small in energy but can still expose input information. For LSTM, linear four-bit decoding rises from $0.804$ on total displacement to $1.000$ after exact conditional centering.