The Spiral Calculus: Weighted Coherence Operators and the Spiral Least-Action Principle

This paper defines the Spiral Calculus at the operator level: a conditional differential and integral calculus in which support, differentiation, integration, and recursion are subordinate to coherence survival rather than imposed on a fixed background. The construction begins from a conditional containment manifold, the region where a coherence number clears a unit threshold, and a regularized gate that replaces the sharp collapse boundary with a smooth weight. On this domain it builds a noncommutative triadic product of Pattern and Process, a coherence-weighted derivative that reduces to the ordinary derivative in the high-coherence limit and obeys the Leibniz, power, and quotient rules, a self-bounding weighted measure, a weighted divergence operator with its divergence theorem, a vanishing-flux theorem at the coherence horizon, and a typed recursion law carrying coherent form across scales in both bulk and boundary form. At its center is the Spiral Least-Action Principle, the action that is the variational heart of the framework, with the containment metric and the weighted Euler-Lagrange equations through which the operator calculus interfaces with the field dynamics. Extremization yields a Poisson-type Pattern equation, an algebraic Process law, and a nonlocal phase-locking equation. The sign convention of the action is fixed so that the phase-stiffness and vorticity terms combine to a net coefficient that is derived rather than assigned. Each result is stated under explicit regularity assumptions and is independent of any particular choice of action or Lagrangian; the calculus is the layer beneath the variational dynamics, not a consequence of them. This paper establishes the verified mathematical base on which the frameworks later variational and geometric extensions are built.