Production deep learning systems across enterprise domains operate under constraints that academic benchmarks routinely obscure: labeled data is expensive, inference budgets are tight, and models that cannot explain their behavior are difficult to trust and maintain. We present ChainzRule (CR), a neural architecture replacing typical activations with learnable polynomial layers governed by Differential Regularization (DREG), a layer-wise Jacobian penalty computed analytically during the forward pass at standard inference cost. The core claim is that bounding intermediate derivatives forces the network toward low-frequency, structurally stable representations, simultaneously reducing dependence on labeled data volume, improving robustness to distribution shift, and providing a measurable, gradient-based handle on model behavior. Evaluated across five domains, CR achieves
85.71%±2.01% on Pima Diabetes (statistically superior to SVM and XGBoost),
46.20%±0.37% on SST-5 sentiment classification with a frozen encoder (superior to RNTN using approximately 5% of its training data),
55.79% on SST-5 with a fine-tuned BERT backbone (versus BERT-base linear head at
54.9%),
70.17% on Yelp Full ordinal regression with 3.2M parameters versus a 10-model average of
66.35%, and
+2.32% mean corruption accuracy on CIFAR-10-C. All results with reported
p-values fall below the
α=0.05 threshold after Bonferroni correction. CR maintains a gradient tail ratio
τ (p99/mean) of
1.01--
1.02 against
1.07--
1.09 for all typical activation function baselines across every data fraction, a structural invariant we propose as the mechanistic driver of sample efficiency and a deployment-time proxy for model reliability.