This is a work in progress. The paper presents the geometric and dimensional implications of the Prime Move Operator — a minimal five-stage cycle (Split → Tension → Failed Merge → Scar → Decay) derived from axioms about distinction and irreversibility. We demonstrate that complexity organizes into a universal ladder of tiers: · Linear (1D)· Hierarchical (2D)· Fractal (fractional dimension)· Wave (3D oscillation)· Dendritic (3D networks)· Perpetual (4D time-binding) Through geometric realization on expanding domains, we derive the emergence of the golden ratio φ ≈ 1.618 and the golden angle ≈ 137.5° from energy minimization. The dimensional progression arises naturally from the operator's recursive structure. Key contributions: · Derivation of golden angle convergence from first principles· Identification of a critical branching point at Tier 3 (Fractal) with three convergent pathways (Spiral, Optimization, Aggregation)· Testable predictions about tier dependencies and pathway signatures· Demonstration across four representative systems (hurricane, circulatory, coastline, consciousness) This is Paper 3 of the Prime Move series. Paper 1 establishes the conceptual foundation (10.5281/zenodo.18998546). Paper 2 provides the mathematical formalization (10.5281/zenodo.19217367). Paper 4 offers empirical validation across fifteen systems. Keywords: Prime Move Operator, hierarchical complexity, dimensional unfolding, golden ratio, geometric realization, scale invariance, Fibonacci sequence