Tangent-Branch Neural Evasion Layer

Use this as a differentiable planning and safety layer between a neural policy and a simulator in multi-agent RL, or as a trajectory-proposal layer for a learned world model. At each timestep, represent every pursuer by a disk center C_i and radius r_e. Let the network output a desired goal point kappa, nominal speed, and residual steering command. For every pursuer i and side q in {-1,+1}, compute d=norm(C_i-kappa); discard the branch if d<=r_e; compute alpha=asin(r_e/d), then S=kappa+sqrt(d^2-r_e^2)/d times R(alpha)^(-q)(C_i-kappa). Estimate the turn-arc duration from the current evader heading and the angular turn-rate limit, estimate the straight duration from norm(S-kappa)/v_e, and set T_b equal to their sum. Select b*=argmin_b T_b, construct the tangent-plus-line reference velocity, and project the network velocity onto the half-space that does not decrease distance to the active disk below r_e. Add the projected residual and clip to the evader speed limit. During training use T_soft=-tau log(sum_b exp(-T_b/tau)) to avoid discontinuous gradients; use hard argmin at inference. The geometric quantities are computed analytically, while turn duration, opponent-motion corrections, and model mismatch can be estimated from rollouts. First test on synthetic 2D episodes with two constant-speed pursuers and random initial states. Compare PPO, PPO plus velocity projection, and PPO plus the tangent layer using escape rate, path length, and samples to success. The predicted signature is a branch switch when two candidate times become equal, and a feasibility transition near v_e/v_p=2, the paper's analytically tractable speed-ratio boundary. The measured transition should lie within 20% of 2, and disk penetration should remain at numerical integration error rather than increasing with policy residual magnitude.