Curved-Critical Residual Lattice

1. Exact integration point: implement the model as a recurrent 2D lattice, a deep-equilibrium layer, or a stack of residual blocks indexed by horizontal position \(j\) and depth/time \(k\). Replace ordinary convolution or recurrent mixing with the four-neighbor update above. Use trainable vectors \(\lambda_j\) and \(\mu_k\), constrain their increments with the smoothness penalty, and fix \(\gamma\) initially. Restrict parameters to a range where the sine weights have consistent signs, or retain absolute values in the normalization. 2. Pseudocode: compute \(q_{jk}=\lambda_j-\mu_k\); set \(a=\sin(q+\gamma)\), \(b=\sin q\), and \(c=\sin\gamma\); normalize by \(s=|a|+|b|+|c|\); update every lattice state with the weighted neighbor sum; add the smoothness and invariant penalties to the task loss. 3. Computed versus estimated: \(a,b,c,\Delta_{jk}\), and \(v_{jk}\) are computed exactly at every step. The continuum metric is an empirical diagnostic; estimate it by injecting impulses at selected lattice sites and fitting the displacement of the largest response over depth. Estimate the largest Jacobian singular value with ten-step power iteration. 4. First cheap experiment: use a 32-layer, 128-token sequence model represented as a \(32\times128\) lattice on permuted-MNIST, comparing a standard residual RNN, an unconstrained inhomogeneous lattice, and this invariant parameterization at equal parameter count. Sweep coupling amplitude and record gradient norms, impulse-response width, validation accuracy, and estimated Jacobian singular values. The predicted signature is a broad near-critical regime where the per-layer Jacobian singular value remains near one and impulse correlations decay much more slowly than exponentially. Unconstrained inhomogeneity should show exploding or frozen propagation earlier. The transition to frozen behavior should occur when the measured local response gain falls below approximately one, while the constrained model should preserve its transition boundary within 20 percent as variation increases until the parameter increments become order one.