Description
This monograph develops a unified theoretical framework centered on the power-law kernel and demonstrates that this seemingly simple functional form generates a rich and universal structure spanning geometry, dynamics, probability, and field theory. The core contribution is the reinterpretation of the kernel not as a mere function, but as a generator of fields. Through systematic analysis, the work establishes a hierarchy of representations: local response functions, global action functionals, dynamical equations of motion, normalized probability distributions, and information-geometric manifolds. These representations are shown to be mathematically equivalent projections of a single underlying structure. A central result is that constraint is fundamentally geometric. The exponent governs the morphology of limitation - determining how curvature is distributed, when nonlinear effects emerge, and how systems transition between regimes of growth, optimality, decline, and collapse. The parameter controls the intensity of limitation, while defines the driving force. Together, these parameters form a signature that uniquely characterizes the system. The monograph further demonstrates that normalization of the kernel yields a generalized Beta-type distribution, enabling empirical interpretation and statistical estimation. Embedding this distribution into a statistical manifold introduces an information-geometric structure, where Fisher information defines a metric and curvature captures variability across systems.