Gradient Fractals - The Fractal Veldt - Essay II: The Algebraic and Computational Ground - The Scale-Invariant Algebraic Tower, the Interaction Formula, and the Computational Syntax of Multi-Node Processing

Essay II of the Gradient Fractals suite executes the Algebraic and Computational layers of the ten-layer derivational chain. These two layers together constitute the arithmetic spine of the Gradient Fractal Field: they establish, with zero free parameters, the exact algebraic structure of multi-node interaction and the exact computational operations that multi-node processing requires. Nothing is imported. No algebraic framework, no computational model, no network theory enters from outside. Every result is forced by the locked constants of the foundational suites and the ontological-logical ground established in GF-I. The Algebraic layer (Part I) establishes three results. First: the Minimal Algebraic Tower T. ACT, established in the foundational suites for the single node, is scale-invariant across every grain δₙ = Nₛₐₜⁿ × δ₀. Each of the four algebraic classes — Rational (R), Rational-Power Arb, Degree-2 Algebraic (A2), Transcendental (T2) — is forced at every grain by the same structural transitions that forced it at δ₀. The algebraic self-similarity of the Gradient Fractal Field is a theorem, not an assumption (T. GF. ACT). Second: the solid angle per Chronon Ω = (E×C) /F² = 14/9 is derived as the invariant angular measure of the single-node worldline sweep on S² (F), and its emergence as the natural unit for multi-node computational density is forced by the Area Sweep Condition and the Registration Sphere geometry. Third: the interaction formula Ωᵈⁿₜ = Ω₁ + Ω₂ − 1 is derived in full from the shared Boundary interface (Deriv. 2. 2, GF-I) and the shared vacuum baseline Nᶛₐᶜ = 10, without importing any external set-theoretic intersection or network-theoretic overlap formula (T. GF. SVB, T. GF. OIN). The N-body generalisation to Ωᵈⁿₜ (N) = Σᵢ Ωᵢ − N (N−1) /2 is derived from the same first principle (T. GF. NBK). The Computational layer (Part II) establishes three results. First: the four irreducible operations Distinguish, Order, Replicate, Self-modify that govern single-node processing are extended to multi-node processing without requiring a fifth operation — inter-node registration is derived to be the Replicate operation at the parent grain (T. GF. OPS). Second: the multi-node parity spectrum — the set of stable N-node configurations determined by the inter-node modular closure condition — is derived as an exact analogue of the single-node parity spectrum k ∈ 3, 9, 15, 21,. . . (T. GF. PAR). Third: the computational density of the Gradient Fractal Field at depth n is derived as Ωₜₒₜₐₗ (n) = Nₛₐₜⁿ × Ω₁ minus the inter-node shared baselines, giving the exact computational capacity of the fractal field at every scale (T. GF. CPX). Throughout, the Nothing-pole (discrete arithmetic) and Something-pole (macroscopic relational structure) are derived simultaneously and with equal necessity. The co-constitutive identity established at each result is not a description but a derivational necessity: the algebraic and computational structure of the Gradient Fractal Field is the same structure viewed from two irreducible poles.