Monodromy Computation Model

Abstract

A closed path can return to its starting point while analytic continuation reaches another branch. We construct a finite computational calculus in which retained logarithmic branches supply memory, analytic observations select operations, and descent through a covering exposes deeper memory. Three branch coordinates suffice, and every finite observation uses the same cubic power covering. Each program defines a globally real analytic return map whose integer-lattice execution is realized by actual path lifting. Our first theorem simulates every finite-alphabet, finite-control Post--Turing machine, preserving complete configurations, halting, and divergence. Our second theorem identifies the partial numeric functions of the calculus, under fixed primitive-recursive input and output maps, with exactly the partial recursive functions. Deleting descent leaves a Forward language with a syntax-derived finite observation quotient and a primitive-recursive halting predicate. This yields a strict separation within the calculus and identifies access to deeper memory as its decisive additional operation. The constructions and semantic correspondences are formalized in Lean 4 with Mathlib.