Forecast Error Cannot Size a Fixed-Point Datapath for Chaotic Time-Series Surrogates

Surrogates of chaotic time series are compressed aggressively for fixed-point hardware, and the compression literature sizes the resulting datapaths by sweeping precision and watching one-step forecast error. We argue this procedure cannot determine a bit budget. Sweeping an echo state network trained on Lorenz-63 across a fixed-point ladder, we find the model's invariant measure—its climate—collapses between 19 and 18 total bits, while one-step RMSE degrades smoothly and monotonically across the entire ladder with no knee, inflection or other feature at the transition. The consequence is not that forecast error is mis-calibrated but that it is unidentifiable: the word length it licenses swings from 22 bits to 14 as the accept threshold varies over a range no practitioner could distinguish a priori, spanning $2.0\times$ to $4.9\times$ in bit-operations, with nothing in the curve indicating which choice is safe. A closed-loop distributional criterion, whose threshold can be calibrated against the metric's own measured sampling-noise floor, places the budget at 19 total bits and $2.66\times$. We identify horizon as the common cause: a short attractor plot and a 6,500-step Lyapunov estimate miss the collapse for the same reason one-step error does. We report an ordering among diagnostics at their conventional horizons, and find that quantization-aware training on a one-step objective reclaims no word length.