When Logic Meets Perception: Operator-Agnostic Differentiable Reasoning for Reliable Neural Prediction

Neural models in high-stakes domains lack access to ontological domain constraints that practitioners take for granted, and retrofitting such knowledge is hard: expressive logical formalisms do not scale, while scalable ones cannot express the negation, disjunction, and quantification that real constraints require. We present a differentiable reasoning framework that resolves this tension. Operating within a decidable logic that retains full Boolean expressivity, it transforms domain rules into a training objective with guaranteed polynomial-time cost per iteration. The framework is operator-agnostic - it decouples logical structure from the choice of underlying continuous semantics, revealing, through the first controlled comparison of its kind, that this choice alone can swing performance by over 30 points on the same task. This finding motivates two adaptive mechanisms: a semantic gate that focuses gradient signal on the model's most flagrant logical violations, and a structure-aware loss that automatically reweights its objective according to the logical complexity of the input constraints. Together, they eliminate the need for per-dataset loss tuning. On eight benchmark ontologies, the framework achieves statistically significant improvements over nine baselines spanning neuro-symbolic, geometric, and probabilistic paradigms. On semantic image interpretation, it refines a frozen object detector using domain rules alone - without the need for extra labels - lifting macro-averaged F1 by up to 7.8%, showing that structured knowledge, properly injected, can turn brittle pattern-matching into logically coherent prediction.