Discrete-Scale Bistable Feature Relaxation

1. Integration point: attach the module after a standard CNN encoder and before the prediction head. The encoder produces an external field $g\in\mathbb{R}^{H\times W}$; initialize $u^0=\tanh(g)$ or a learned affine transform of $g$. Run 8–32 relaxation steps, returning both the final continuous state $u^T$ and the binary map $s^T$. Use a 4- or 8-neighbor adjacency matrix for $A$. 2. Pseudocode: compute $g=\mathrm{CNN}(x)$; set $u=\tanh(g)$; for $n=1,\ldots,T$, compute $Lu$ by depthwise convolution, set $u\leftarrow u+\eta[u-u^3+\kappa Lu+g]$, and record the wall count and number of sites whose sign changed. Define a cascade when the sign-change count exceeds the running median plus $q$ median absolute deviations; optionally skip or compress intermediate steps after a cascade. Train with task loss on $u^T$ plus $\lambda_b\sum_i(u_i^2-1)^2$ and a small total-variation penalty. 3. Computed versus estimated: the reaction, Laplacian, wall count, and cascade detector are computed exactly; $\delta$, $\Lambda$, $B$, and $\phi$ are estimated from validation trajectories by fitting $\log\rho$ against $\log t$ and the residual against $\log t$. Do not impose log-periodicity initially; use it only as a diagnostic or weak regularizer. 4. First cheap experiment: train a small U-Net on Fashion-MNIST or a small binary segmentation dataset, comparing ordinary per-pixel logits with the bistable module at matched FLOPs. Sweep $\eta$ and $\kappa$. The predicted signature is a parameter band with long low-activity plateaus followed by sign-change bursts, geometric cascade times with $R^2>0.8$ for $\log t_k$ versus $k$, and a fitted exponent $\delta<0.5$ with a significant periodic residual in $\log t$. Outside this band, dynamics should become either smooth or unstable; the measured transition should be reproducible across seeds.