Proper-Kernel Neural Safety Layer

1. Integration point: use a differentiable MLP or recurrent policy $\pi_\theta(o_t)$ in a simulated robot or low-dimensional continuous-control environment. At every control step, estimate $h(x_t)$ and its gradient, maintain one scalar filter state $z_t$ per safety constraint, and solve a projection QP that finds the smallest correction $\Delta u_t$ to the policy action. 2. Pseudocode: compute $u_\pi=\pi_\theta(o_t)$; estimate $q_\pi=\nabla h(x_t)^T[f(x_t)+g(x_t)u_\pi]+\alpha h(x_t)$; update $z\leftarrow e^{-a\Delta t}z+(1-e^{-a\Delta t})q_\pi$; solve $\min_{u}\|u-u_\pi\|_2^2+\rho\|s\|^2$ subject to $\nabla h^Tg u\geq-\nabla h^Tf-\alpha h+(1-\kappa/a)z+\varepsilon-s$, $h\geq\delta-s$, and $\|u\|_\infty\leq u_{\max}$. Apply $u$ and update the filter with the executed residual. 3. Computed from the paper's mechanism: the exponential proper kernel, dynamic barrier state, affine-QP structure, and robust error margin. Estimated empirically: $f,g$, the observation-noise bound $\varepsilon$, and the useful bandwidth $a$. 4. First experiment: use third-order double-integrator-like dynamics with noisy state observations, comparing equal-network and equal-control-rate baselines: an unconstrained policy, an ordinary memoryless CBF-QP, and the proposed filtered CBF-QP. Sweep $a\in\{0.1,0.3,1,3,10\}$ and noise amplitude while holding training steps and action limits fixed. 5. Preregistered signature: for sinusoidal barrier-residual noise at frequency $\omega$, measured residual amplitude must follow $a/\sqrt{a^2+\omega^2}$ within 20%, and action total variation must fall by at least 50% when $\omega\geq5a$. The safety violation rate must remain below the calibrated error-bound rate of the memoryless robust baseline. Reject the transfer if the frequency response misses 20% or if filtered safety-QP infeasibility increases by more than 5% relative to the ordinary CBF-QP.