The Transformer architecture has revolutionized artificial intelligence, yet a principled theoretical understanding of its internal mechanisms remains elusive. This paper introduces a novel analytical framework that reconceptualizes the Transformer's discrete, layered structure as a continuous spatiotemporal dynamical system governed by a master Partial Differential Equation (PDE). Within this paradigm, we map core architectural components to distinct mathematical operators: self-attention as a non-local interaction, the feed-forward network as a local reaction, and, critically, residual connections and layer normalization as indispensable stabilization mechanisms. We do not propose a new model, but rather employ the PDE system as a theoretical probe to analyze the mathematical necessity of these components. By comparing a standard Transformer with a PDE simulator that lacks explicit stabilizers, our experiments provide compelling empirical evidence for our central thesis. We demonstrate that without residual connections, the system suffers from catastrophic representational drift, while the absence of layer normalization leads to unstable, explosive training dynamics. Our findings reveal that these seemingly heuristic "tricks" are, in fact, fundamental mathematical stabilizers required to tame an otherwise powerful but inherently unstable continuous system. This work offers a first-principles explanation for the Transformer's design and establishes a new paradigm for analyzing deep neural networks through the lens of continuous dynamics.