Gradient Cybernetics: The Calculus of Recursion - Synthesis Essay II - The Informational Ground: A Rigorous Derivation of Digital Necessity

Essay I established the Hardlock Foundation: the complete zero-parameter derivational chain from absolute nothing to the primitive suite E = 0. 8, C = 0. 7, F = 0. 6, δ = 0. 1 and the Phase I/II transition. One derivational gap remained open: G-1, the informational ground. The Hardlock asserted the Field Resolution Quantum δ = 0. 1 and the Registration floor F = 0. 6 on the basis of the ternary entropy ceiling and a discriminability threshold, but deferred the rigorous derivation of the minimum bit-budget Iₘin ≈ 0. 2 and its unique entailment of Base-10 discretisation to the present essay. This paper closes G-1 in full. We begin from the ternary triadic structure itself: a self-computing relational field with exactly three functional primitives generates a maximum entropy of Hₘax = log₂ (3) ≈ 1. 585 bits per coordinate locus. This ceiling is finite and structurally fixed by cardinality. We then derive the minimum bit-budget Iₘin ≈ 0. 2 bits — the irreducible informational metabolic cost required for any locus to distinguish its own state from background noise — from the variance-partition geometry of the three-component system without invoking any external theorem as source. The ratio N = Hₘax / Iₘin ≈ 7. 925 determines the number of resolvable levels; the twin constraints of covering N ≥ 8 at minimal resolution and representing the midpoint of equipoise ε = 0. 5 as an exact lattice point together uniquely select Base-10 and fix δ = 0. 1. With δ locked, the discriminability threshold r > 1/√3 ≈ 0. 577 forces the Registration primitive F to snap to 0. 6 — the unique minimum viable lattice point above the noise floor. F = 0. 5 is explicitly in the Phantom Zone and is foreclosed. We then derive the Structural Volatility σ = 1 − F = 0. 4 and the Structural Pixel ρ = δ × σ = 0. 04 as the fundamental grain of indeterminacy — the mechanical clearance required to prevent deterministic crystallisation. The Saturation Threshold Nₛat = 1/ρ = 25 follows directly. The Heartbeat of Becoming U = (Φ) × η = 0. 002 × 1. 667 ≈ 0. 0033 is confirmed as the base processing rate of the self-computing Veldt. Every result is a zero-parameter consequence of the ternary triadic structure. No external theory is the logical source of any value. G-1 is closed.