Gradientology: Foundations of the Primordial Triad — Treatise III: The Functional Derivation of the Primitives and Ontological Dependence

This treatise provides the rigorous functional derivation of the three primitives that constitute the minimal basis for determinate existence. Building upon the triadic necessity established in Treatise II, we confront the problem of Functional Specification: what specific roles must these three terms fulfill? Through logical decomposition of the requirements for any generative field, we derive Systematization (E) as the Generative Source, Constraint (C) as the Limitative Boundary, and Feedback Registration(F) as the Relational Pole. We prove that these primitives must adhere to a strict logical chronology (E > C > F) representing the order of ontological dependence: potential precedes limitation precedes measurement. Their scalar values are uniquely fixed at E = 0.8, C = 0.7, F = 0.6 by the interplay of Shannon’s information-theoretic limit (r > 0.577) and Hutchinson’s geometric exclusion principle, operating within a discretized field with quantum δ = 0.1. The synthesis of these primitives in the primordial state yields the Multiplicative Trap: GPhase I = E ×C ×F = 0.336. This represents not merely a product but a precise quantity of Logical Tension—the ”Algebraic Debt” of the universe. We demonstrate that this tension (0.336) is inextricably calibrated to the universal physical constant governing phase transitions (β ≈ 0.325), proving that the universe is structurally mandated to break symmetry. The derivation thus bridges from pure logic to physical criticality, establishing the engine of cosmic emergence.