The Spiral Generator Operator and the Architecture of Recursive Coherent Form

This paper develops the operator-level architecture of the cross-scale structural law of coherent form F = PQR. The structural law states what a persistent system is at a single scale: a product of three jointly necessary roles, Pattern P, Process Q, and Rhythm R, that must clear a critical coherence threshold. What it does not supply is the dynamical object that carries coherent form from one organizational scale to the next, together with the recursion that object generates over nested scales. This paper supplies both. The central construction is the Spiral Generator Operator Sn = An * exp(i * psi_n), in which the amplitude An = PnQnrn bundles Pattern, Process, and local phase coherence into a real non-negative magnitude, and the phase psi_n carries the cross-scale lock that binds a scale to its parent and child. This encoding rests on a decomposition of Rhythm into two independently failing components, local phase coherence and cross-scale lock, and it corrects an earlier operator formulation that placed all of Rhythm in the phase and so lost the Rhythm magnitude from the operator modulus. The construction yields three results: (1) A consistency theorem: |Sn| = PnQnRn = Fn, so the operator modulus reproduces the node-level form exactly and discards nothing. (2) An operator-level threshold: the Spiral Coherence Number SCn = inf_x |Sn| / delta_c, identical in form to the node-level SC = F/c, whose unit value is the marginal-stability fixed point of the recursion rather than a choice of units. (3) A clean separation of two failure modes, local decoherence and cross-scale unlock, each represented by a distinct part of the operator. The discrete operator chain is shown to be the coarse-grained image of a continuous spiral field whose evolution is governed by a constrained variational principle. The architecture therefore has three layers: the law specifies what exists, the field specifies how it evolves, and the operator chain specifies how recursion amplifies or terminates form across scales.