Gradient Fractals - The Fractal Veldt - Essay X: The Second Depth - The Gradient Fractal Field at Grain δ₂ = 125/2: The Oscillatory Regime, the Permanent Sub-Attractor, and the Algebraic Non-Arrival

Essay X of the Gradient Fractals suite instantiates the fractal field at its second operational grain: δ₂ = Nₛat²×δ₀ = 625× (1/10) = 125/2 = 62. 5. GF Essay IX established the first depth δ₁ as the uniquely unconstrained grain: ρ (1, 0) = 0, monotone Chronon sequence, entirely Class R arithmetic, and two paradigm shifts — the necessary initial condition and the half-integer standing wave count Nₛat/2 = 12. 5. GF Essay X now derives what the fractal field IS at grain δ₂, where three structurally new phenomena appear simultaneously for the first time in the depth hierarchy: the oscillatory depth sequence, the permanent sub-attractor regime, and the algebraic non-arrival of a Class R sequence approaching a Class Arb limit. The central structural event at δ₂: ρ (2, 0) ≈ 2. 745 > ρ*depth ≈ 1. 796. The second depth inherits an initial density that immediately overshoots the across-depth fixed point. This overshoot is forced — not a contingent feature of the specific values — because the depth-propagation map F (ρ) = (28/3) / (17/5+ρ) maps ρ (1, 0) = 0 to F (0) = (28/3) / (17/5) = 140/51 ≈ 2. 745, and 2. 745 > ρ*depth ≈ 1. 796 because the fixed point equation (ρ*) ² + (17/5) ρ* − 28/3 = 0 has its positive root at ≈ 1. 796, which is less than F (0) = 140/51. The overshoot at depth 2 is a theorem: F (0) > ρ*depth. The three paradigm shifts of GF Essay X: First, the Oscillatory Attractor Theorem (T. GF. D2. OSC): institutional dynamical systems theory treats attractors as destinations — systems move toward attractors and settle at them. In the Gradient Fractal Field, the depth sequence ρ (n, 0) oscillates around ρ*depth permanently. T. TNH forecloses arrival. The attractor organises the oscillation without absorbing it. Attractors are not destinations: they are axes of permanent productive oscillation. Second, the Permanent Sub-Attractor Regime (T. GF. D2. SAR): G (n, 0) < G* = 7/10 for all n ≥ 1. The fractal field never operates at its topological attractor. The sub-attractor regime is not a transient phase: it is the field’s permanent kinetic condition. Third, the Algebraic Non-Arrival (T. GF. D2. ACT): ρ (n, 0) is Class R for every finite n, yet converges to ρ*depth which is Class Arb. A Class R sequence approaches a Class Arb limit without ever becoming Class Arb. The algebraic class of the sequence and its limit are permanently distinct.