The Spiral Field Dynamics Explicit Coupled Field Equations of the Spiral Least-Action Principle

This paper derives the explicit coupled field equations of the Spiral Least-Action Principle and presents them as a closed system. Beginning from the canonical Spiral action and the gate-weighted Euler-Lagrange theorem of the Spiral Calculus, variation with respect to the four fields, Pattern, Process, phase, and the amplitude that carries the coherence gate, yields: a gate-weighted Poisson equation for Pattern with a coherence-gradient drift term, an algebraic constitutive law for Process, a nonlocal phase-locking equation in which phase curvature and vorticity are sourced by the Pattern Process product through the rhythmic response, and an amplitude relation that is algebraic rather than propagating, because the canonical action carries no amplitude-gradient term, together with the gate feedback that the amplitude exerts on the whole Lagrangian. The phase-stiffness and vorticity terms enter with opposite signs, so the phase-gradient sector carries the net coefficient derived in the framework rather than an assigned one, and the amplitude equation fixes the gauge lock between the phase gradient and the vector potential that grounds the force-free equilibrium. Imposing the operator-closure constraint that binds the amplitude to the Spiral Product of Pattern and Process, through a Lagrange multiplier, gives the constrained system; the multiplier is algebraic in the fixed-metric sector and acquires dynamics only under full metric variation. In the high-coherence limit the gate weight tends to unity and the system reduces to the ordinary Euler-Lagrange field equations of the action. Every equation is derived from the published canonical action and was verified by symbolic computation.