Robustness-capacity feasibility controller

Integrate this into training of a two-layer ReLU MLP immediately after each validation interval. Use a minibatch or held-out probe set to estimate the current fitting error $\hat\varepsilon$ and the input Lipschitz constant. Estimate $\operatorname{Lip}(f)$ by power iteration on the input Jacobian, $L_{\mathrm{est}}=\max_{x\in B}\|J_f(x)\|_2$, using 32--128 random probe points plus training examples; for scalar output, the Jacobian is a row vector. Estimate the noise floor from repeated labels, a known corruption rate, or a robust residual variance estimator. Compute $L_{\min}=c_0\hat\varepsilon\sqrt{n/[(m+1)\log(C_0(m+1)nd/\max(\hat\varepsilon,\delta))]}$, with conservative tunable constants $c_0,C_0$ and numerical floor $\delta$. Pseudocode: (1) train normally for an interval; (2) measure $\hat\varepsilon$ and $L_{\mathrm{est}}$; (3) if $\hat\varepsilon$ is below the noise estimate and $L_{\mathrm{est}}<\eta L_{\min}$, either stop reducing the fit loss and increase a Jacobian penalty, or widen the hidden layer by 25--50%; (4) copy existing neurons and initialize new neurons with small output coefficients; (5) continue training. The theorem supplies the scaling law; $c_0,C_0$, the Jacobian estimator, and the widening threshold are empirical. First test on noisy two-moons, CIFAR-10 binary subsets, and synthetic spherical data using a width-swept two-layer ReLU baseline. Success is lower validation error or better adversarial/noise robustness at matched parameter-update FLOPs, together with fewer late-stage epochs spent driving training loss below the noise floor.