Introduction

Introduction

A recurring theme in positive propagation and transfer-matrix models is that quantitative regularity of the induced projective dynamics is governed not only by positivity but by a finite collection of structural and recurrence features. This paper formalizes that principle in a reduced setting where “locality” produces a finite template-with-bounds interface and an associated finite variational object.

The goal is not an absolute “locality forces spectral gap” theorem. Counterexamples exist without additional mixing/positivity hypotheses. Instead, we prove the strongest honest endpoint available in this architecture: a complete obstruction-theoretic boundary characterization of uniform minimizer-facing projective regularity (PR), together with a strict genericity theorem showing that a rigid, periodic minimizing core is prevalent in the fixed reduced parameterization.

What is meant by PR on minimizers.

Given a nonnegative matrix cocycle acting on the positive cone, PR refers to uniform projective contraction (e.g.\ in the Hilbert metric) along the minimizing set of trajectories/controls selected by the reduced variational problem. The main equivalence theorem identifies a finite set of certifiable “contract gates” whose conjunction is necessary and sufficient for such uniform contraction.

How cycle-core enters.

The locality reduction yields a finite directed graph $G=(V,E)$ and an additive cost $c$ on edges. The cycle-core is the union of minimum-mean cycles. Minimizing invariant measures are supported on this set, so minimizer-facing statements reduce to core-facing statements.

Genericity.

Within the fixed reduced parameter space (edge-cost coordinates $c\in\mathbb R^E$), ties between distinct cycle means lie on proper affine hyperplanes. Hence unique minimizing cycle structure is open–dense and full measure. This prevalence result collapses recurrence on the minimizing set to a periodic phenomenon on a full-measure chamber set.

Constructiveness.

A canonical witness family is synthesized directly from the cycle-core SCC decomposition, removing hand-chosen witness dependence at the covering stage. The remaining nontrivial boundary burden is a certified projective contraction event (Doeblin/primitive block) and excursion-safety.

Related literature.

The projective-metric approach to positive operators goes back to Birkhoff’s contraction coefficient and subsequent refinements (see, e.g., [birkhoff1957extensions,bushell1973hilbert]). Transfer-matrix and subadditive-ergodic perspectives connect to random matrix products and Lyapunov exponents [furstenberg1960products,bougerol1985products]. The symbolic dynamics viewpoint (subshifts, cocycles, minimizing measures) underlies the cycle-core formalism [walters1982introduction].

Organization.

Section fixes the reduced objects and PR notion. Section states the Master Reduction Theorem. Section states the Cycle-Core Universality (boundary equivalence) theorem and derives the minimizer lift and PR rate bounds as corollaries. Section gives strict genericity. Section gives constructive witness synthesis. Appendix material contains the certification interface, constants, obstruction certificates, and proof skeletons.