SCULPT: Backpropagation-Free Linear-Head Adaptation with Symbolic Constraints

Non-differentiable solvers and rule engines provide structured feedback on neural predictions, yet translating this feedback into lightweight post-deployment updates remains challenging. We introduce Symbolic Constraint Updates for Lightweight Predictor Tuning (SCULPT), a backpropagation-free framework for adapting linear prediction heads over frozen representations. The key observation is that a linear inequality on prediction scores maps exactly to a half-space in head-parameter space, providing a directly computable update direction. To accommodate feedback with varying reliability and structure, SCULPT combines reliability-gated directions from violated constraints with covariance-aware fitting of consensus-filtered solver targets. A shared feedback router supports either feedback type or their combination, with an optional trust projection for drift control. Experiments on routing, visual Sudoku, and corrupted digit addition demonstrate the utility of this interface, including a gain of 6.44 percentage points in Sudoku cell accuracy over the supervised head. \method provides a unified approach to incorporating actionable symbolic feedback into deployed predictors without retraining their representations or differentiating through the checker.