The Spiral Engine describes the unified mechanism by which self-referential coherence converts between its two forms: propagating (electromagnetic) and standing (mechanical/mass). The spiral closes in exactly 137 turns — the winding number of the proton. The escape fraction 1/φ propagates as light; the capture fraction 1/φ² freezes as mass. The ratio of capture to escape is φ⁻¹, establishing E = mc² as the exchange rate between the two states. The engine's 137-turn coprimality lock explains why electromagnetism and gravity are long-range (they complete the closure) while the other forces do not (they freeze before closing).