Scaled Reciprocal Safety Layer

1. Integration point: use this as a differentiable safety layer after a neural policy aπ = πφ(z), or as a regularizer during model-based reinforcement learning and neural-ODE training. The model must expose f(z) and g(z), either from known physics, a learned control-affine model, or local Jacobian estimates. Choose a differentiable safety margin h₀(z), such as squared distance to an obstacle minus a radius, an actuator-temperature margin, or a latent-state operating envelope. 2. Pseudocode: compute h = h₀(z), q = gradient(h) transpose f(z), s = 1 + δ tanh(κq), B = s/(h + ε), Lᵍq = gradient(q) transpose g(z), and LᵍB = δκ sech²(κq)Lᵍq/(h + ε). Compute LᶠB by automatic differentiation. Project aπ onto the smallest correction satisfying LᶠB + LᵍB a ≤ γB and action bounds; alternatively add λ max(0, LᶠB + LᵍB aπ − γB)² to the training loss. 3. Computed versus estimated: h, gradients, and Lie derivatives are computed by autodiff when f, g, and h₀ are available. Otherwise estimate f and g from short action-conditioned trajectories and validate derivative estimates with finite differences. Clip B only for numerical overflow, not near the evaluation boundary. 4. First cheap experiment: train a small MLP policy on a two-dimensional point-mass or unicycle obstacle-avoidance task using PPO or behavior cloning, comparing no barrier, ordinary 1/h₀, and the scaled barrier. Sweep action-step size and evaluate one-step constraint feasibility. The quantitative prediction is that ordinary 1/h₀ has LᵍB = 0 when Lᵍh₀ = 0, whereas the scaled barrier has LᵍB approximately equal to δκLᵍq/h₀ at q = 0. Measure the fraction of states where the safety projection is feasible and verify that the scaled method retains nonzero action authority near zero radial velocity. Under an accurately learned model, trajectories satisfying the inequality should show zero boundary crossings; robustness should degrade when derivative-estimation error exceeds the available slack γB − (LᶠB + LᵍB aπ).