ChainzRule: Sample-Efficient, Robust Deep Learning Across Tabular, NLP, and Vision Tasks

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.