We propose a foundational route from elementary mathematical operations to the structural form of physical law. The guiding thesis is that multiplication is the primitive operation that generates geometric extension (e.g., area via bilinear composition), integration is the continuous accumulation of such local extensions into global quantities, and differentiation (or functional variation) is the dual operation that extracts local constraints from global accumulations. From these principles, we show how any consistent description of ``physical reality'' must be formulated in terms of local densities defined over a continuous geometric support, whose global content is obtained by integration and whose dynamics follows from variational (action) stationarity.Within this operational framework, quadratic field terms arise naturally as the simplest scalar invariants built from local degrees of freedom, while source couplings appear as bilinear products between generalized currents and the underlying deformation variables. Furthermore, we show that quantum entanglement is not a dynamical anomaly but a structural inevitability: additive accumulation acting on states represented in a multiplicative (spectral) basis generically produces global correlations that resist local factorization. This reframes Bell-type violations as a failure of structural independence rather than a signal of superluminal causal influence, thereby preserving relativistic causality at the level of dynamical propagation.Crucially, beyond the contractive modes commonly associated with forces and curvature, the same logic compels expansive degrees of freedom: an entropic sector characterized by an intensive--extensive product structure (temperature-like $times$ entropy-like) contributing intrinsically to the global action. This viewpoint yields a general blueprint for interpreting electromagnetic, gravitational, and entropic responses as projected modes of a common underlying field structure, and it clarifies why concrete realizations of such a blueprint---including quantum-elastic and gravito-entropic field models---arise as minimal, structurally stable completions rather than independent hypotheses.