Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones

Abstract

Alternating reciprocal transformations of polynomial divisibility systems couple an orbit of integer points to a rational dynamical system on polynomial coefficients. For fixed positive degrees $n,m$, we prove universal four-step return to rational gauge classes. Weighted primitive representatives preserve integral continuation through nondegenerate jumps using only exact division and coefficient gcds; the stronger requirement of monic integral representatives is governed by endpoint $p$-adic valuation defects. After one integral block, constant-term valuations evolve by an explicit unimodular matrix whose trace depends only on $nm$. This gives finite-order behavior for $nm<4$, parabolic dilations for $nm=4$, and hyperbolic dynamics for $nm>4$. In the hyperbolic case, invariant thresholds, irrational eigencones, and discrete valley formulas give finite criteria for forward and bi-infinite integral coefficient orbits, although failure can be delayed arbitrarily long. Every positive-constant canonical monic coefficient chain admits an explicit positive rational four-phase gauge. At $nm=4$, geometric gauges reduce nondegenerate compatible rational point orbits to two-periodic affine recurrences, hence to sequences satisfying linear recurrences with constant coefficients. Polynomial identities of degree at most $h$ on each invariant conic admit certificates from $2h+1$ initial phase states, while integral unimodular recurrence models yield periodic modular obstructions. For $nm>4$, eventually positive normalized point orbits tending to infinity have hyperbolic logarithmic growth and cannot satisfy such linear recurrences. Finally, separated auxiliary-relation transport reduces rational rank to at most one after two alternating jumps; quadratic specialization gives exact local saturation criteria and persistence results across periodic, parabolic, and hyperbolic orbits.