Large Deep Neural Networks (DNNs) deliver state-of-the-art accuracy on vision and language tasks yet remain impractical for deployment on edge and resource-constrained devices. Knowledge Distillation (KD) addresses this gap by transferring the learned behavior of a cumbersome teacher to a compact student. Although the seminal temperature-scaled formulation of Hinton et al. [1] has inspired many extensions, the theoretical reasons why small students can faithfully mimic large teachers remain only partially understood, and existing distillation pipelines rarely exploit the geometric structure of the teacher's decision map. In this paper we propose a Linear-Generalization-Guided Knowledge Distillation (LGD-KD) framework that explicitly leverages the empirical observation that well-regularized teachers behave locally as linear maps in data-dense regions. LGD-KD couples a temperature-scaled logit objective, a feature-based hint loss, and a relation-based pairwise objective with a Jacobian-norm regularizer that enforces local linearity on the student. All design stages are presented as formula tables that disambiguate every symbol, including the frequently misread subscript T in fₜ (the teacher function) and the F symbol (the generic function family). Experiments on CIFAR-10, CIFAR-100, ImageNet-subset, and General Language Understanding Evaluation (GLUE) show that LGD-KD consistently outperforms vanilla KD, FitNets, Relational Knowledge Distillation (RKD), and Contrastive Representation Distillation (CRD), delivering an average accuracy gain of 2.61 percentage points over the strongest baseline. Paired Wilcoxon signed-rank tests across ten random seeds confirm statistical significance (p < 0.01) in every setting, and the resulting students consume 7.8× fewer FLOPs and 11.4× fewer parameters than the teacher while retaining 95.2% of its accuracy. The paper concludes with a rigorous peer-review by five referees and an editor's revision report, demonstrating that the manuscript meets the methodological and reporting standards of Expert Systems with Applications.