Tiny Local Recurrence with Adaptive Computation

Integrate this as a replacement for a conventional stack of Transformer feed-forward blocks or residual blocks, leaving the surrounding embedding and output head unchanged. Use a latent tensor $s\in\mathbb R^{C\times H\times W}$ for the simplest experiment, or $s\in\mathbb R^d$ for a vector version. The encoder writes the token or example representation into designated channels or positions, and the decoder reads the final state. Define one shared transition as s = s + alpha * local_rule(norm(s)), where local_rule is a depthwise 3x3 convolution followed by a pointwise two-layer MLP; all weights are reused at every microstep. Pseudocode: s=E(x); mass=0; state_acc=0; for t in range(Tmax): s=f(s); h=sigmoid(wh@pool(s)+bh); delta=min(h,1-mass); state_acc += delta*s; mass += delta; if mass >= 1-eps: break; y=D(state_acc + eps*s). Compute the task loss plus the ponder penalty $\lambda T$ shown above. The paper contributes the fixed-rule recurrence, interface separation, and adaptive $T$ interpretation; the halting probability and penalty are an explicit engineering adaptation. First test on MNIST, CIFAR-10, and a synthetic arithmetic extrapolation task using a 6-layer residual MLP baseline with matched parameter count. Measure validation accuracy, average microsteps, wall-clock time, FLOPs, and extrapolation accuracy outside the training range. Success means equal accuracy with at least 2x fewer learned parameters, or lower average computation on easy examples without a worse hard-example error curve.