Gradientology: Foundations of the Primordial Triad — Treatise VIII: the Information-Theoretic Derivation of Registration and the Digital Necessity

This treatise inaugurates the Quantification phase of Gradientology by deriving the specific scalar values of the primordial triad from the first principles of Claude Shannon’s Information Theory. We model the Relational Field as a stochastic communication channel with three degrees of freedom. By applying probability theory, we derive the Statistical Floor of existence: the minimum correlation (rcrit ≈ 0.577) required for a triadic signal to be distinguishable from entropic noise. TheShannon-Hartley theorem then mandates the discretization of the field, yielding a Field Resolution Quantum of δ = 0.1 via a Base-10 lattice. The Principle of Informational Quantization (The Snap) forces the continuous threshold to ”snap up” to the nearest valid lattice point, uniquely fixing the Registration primitive at F = 0.6. Applying the Principle of Minimal Differentiation to the functional hierarchy (E > C > F ) then fixes C = 0.7 and E = 0.8. We prove the scalar suite {0.8, 0.7, 0.6} is the unique solution that simultaneously satisfies the noise floor, hierarchy, and criticality condition (T I = 0.336 ≈ β = 0.325). This convergence demonstrates that the fundamental constants of reality are not arbitrary but are the inevitable result of a system solving its own existential equation under informational constraints.