Singular Gradient-Barrier Continuation

1. Integration point: apply the barrier to the input Jacobian of a scalar network output f_theta(x), such as an MLP used for a signed-distance field, coordinate neural field, energy model, or critic. For vector outputs y_theta(x), either sum the barrier over output coordinates or estimate the largest singular value of the Jacobian with one-step power iteration. The barrier is evaluated during training on minibatch inputs, not at inference. 2. Pseudocode: sample x from the training batch; enable gradients on x; compute y=f_theta(x); compute g=autograd.grad(y.sum(), x, create_graph=True); set q=(g.square().sum(dim=-1))/G**2; set z=clamp(1-q, min=delta); set barrier=(-log(z)/beta).mean(); set loss=task_loss+lambda*barrier; backpropagate loss and update theta. Start with beta=0.1 and lambda=lambda_0, then every K steps multiply beta by 1.5 or multiply lambda by 1.5, while keeping q below 1-delta. 3. The mathematical quantities are s=q/2, the singular derivative b_beta=1/(1-q), and the target strict margin q<= (1-epsilon)^2. The empirical quantities are minibatch gradients, the fraction of samples with q>1, the 95th and 99th percentiles of gradient norms, and the largest observed norm on a held-out probe grid. Use gradient clipping only as an emergency fallback when q exceeds 1-delta; do not rely on clipping as the main mechanism. 4. First experiment: train a 4-layer width-256 tanh MLP on a 2D analytic regression function with a known Lipschitz constant, comparing ordinary MSE, MSE plus squared gradient penalty, and the singular continuation barrier at matched training FLOPs. Then test a small neural signed-distance-field model on a 2D or 3D shape. Measure task error, maximum and 99th-percentile input gradient, number of boundary violations, training NaNs, and wall-clock overhead. Success means substantially fewer violations and a lower worst-case gradient at comparable task error, with smoother loss curves and no increase in instability; a stronger result is equal gradient safety with less penalty weight or faster convergence than squared-gradient regularization.