L1 Priority Penalties with Squared-Hinge Escape Detection

Integrate this at the loss layer wherever objectives have strict ordering, for example safety before legality before comfort in trajectory prediction, validity before likelihood in structured generation, or toxicity before style. For each minibatch compute g_{i,j}(f_theta(x)), r_i=sum_j relu(g_{i,j}), and loss=J+sum_i w_i r_i. Backpropagate this L1 loss with AdamW or SGD. For comparison, train an otherwise identical squared-hinge model using sum_j relu(g_{i,j})^2. Every K steps, log each pair (w_i,r_i), the norm of the tier gradient, and the lower-tier objective. Over a window of M checkpoints, fit the slope of log(r_i+tau) against log(w_i+tau); a slope near -1 indicates the predicted O(1/w) penalty regime. If r_i remains above a required tolerance while increasing w_i gives lower-tier improvement and the high-tier gradient norm is small, do not keep multiplying w_i. Instead execute a fallback for B steps: optimize only V_i, or project the model output onto g_{i,j}<=0 when a differentiable projection is available, then resume the weighted loss. The mathematical quantities are the L1/L2 subgradients, residual, and inverse-weight diagnostic; slope thresholds and tolerances are empirical. Test first on a small MLP or transformer with three synthetic ordered classification constraints against equal weights, fixed 100:10:1 weights, and sequential fine-tuning. Report high-tier violation rate, lower-tier accuracy, wall-clock cost, and residual versus weight. Success is finite-weight near-zero violations for L1, a visible inverse-weight tail for squared hinges, and fewer high-tier violations after fallback.