Distinct-kink complexity regularizer and merger

Apply this at optimizer checkpoints or before inference for a two-layer ReLU network. For every nonzero hidden weight, compute $u=w/\|w\|_2$, $\beta=b/\|w\|_2$, and $\alpha=a\|w\|_2$. Canonicalization has a sign subtlety: retain a fixed orientation convention for $(u,\beta)$, and only merge entries whose full normalized pair agrees within tolerance; do not merge $(u,\beta)$ with $(-u,-\beta)$ unless the corresponding affine correction is explicitly derived. Hash rounded $(u,\beta)$ vectors or use locality-sensitive nearest-neighbor search with tolerance $\tau$. For each cluster, replace all members by one neuron with coefficient $\alpha_{\mathrm{cluster}}=\sum_i\alpha_i$, preserving the shared $(u,\beta)$; discard coefficients below a threshold only in approximate compression mode. During training, add $\lambda_k\tilde{k}$ to the loss, where $\tilde{k}=\sum_{p\in\text{clusters}}\operatorname{softplus}((\|\alpha_p\|-\alpha_0)/s)$ is a differentiable proxy, and periodically perform exact discrete merging. Re-estimate $k(f)$ after every merge and log the theorem-inspired lower bound using $k(f)+1$, not raw width. The exact algebraic rescaling and coefficient summation come from the canonical kink representation; clustering tolerance, penalty weights, and pruning thresholds are empirical. First test on noisy 2D regression and CIFAR-10 with a width-1024 shallow ReLU model, comparing raw-$L_0$ pruning and magnitude pruning. Measure exact function discrepancy on a dense probe set, parameter reduction, validation accuracy, Jacobian norm, and robustness to input perturbations. Success means at least 2x fewer hidden units at near-identical predictions, with lower or unchanged Jacobian norm and no worse validation error.