The Prime Move Hierarchy: Dimensional Unfolding and Tier Emergence (Paper 3)

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