Monotone Ising-Symmetric RG Pooling

Integrate this at every stride-2 downsampling point in a CNN, or at every graph coarsening step for a graph neural network. First convert each scalar feature to a bounded spin using s_i=tanh(x_i/ au); for a four-site spatial block use hard signs during evaluation or a straight-through sign estimator during training. Compute k=\sum_i\mathbf{1}[s_i>0], look up (p_0,\ldots,p_4)=(0,\alpha,1/2,1-\alpha,1), and produce a coarse spin by sampling z\sim Uniform(0,1) and setting ilde{s}=+1 if z<p_k, otherwise -1. Multiply the sampled spin by a learned positive amplitude a, or use it as a gate on the block average, y=a ilde{s}(\frac14\sum_i|x_i|). Parameterize \alpha=\frac12\sigma(\beta) so its required range is automatic. For lower-variance training, replace sampling with the expectation E[ ilde{s}\mid k]=2p_k-1; optionally use Gumbel noise for a relaxed stochastic version. The exact mathematical constraints are normalization, sign equivariance, and monotonicity; no paper-specific quantity needs to be estimated. Compare against max pooling, average pooling, and learned strided convolution on CIFAR-10 or ImageNet-32 using an equally sized ResNet. Test fixed \alpha\in\{0,0.25,0.5\} versus learned \alpha, with and without stochastic sampling. Measure validation accuracy, sensitivity to additive noise and global sign flips, gradient variance, and activation memory. Success means equal-or-better accuracy than average pooling, improved perturbation robustness, and no more than a small training slowdown; in a world-model setting, also test whether long rollouts preserve stable coarse sign patterns.