A Unified Framework for Osculation-Based Divisibility Testing in Arbitrary Positional Numeral Systems

Divisibility rules are traditionally presented as collections of base-specific arithmeticprocedures, often derived independently for particular divisors and numeral systems.Although many such rules can be justified through modular arithmetic, existing treatments remain largely focused on isolated constructions and provide limited insight intothe general structural principles underlying recursive digit-reduction methods.This paper develops a base-independent theory of osculation for positional numeralsystems. Let b denote an arbitrary base and let m be an integer satisfying gcd(b, m) = 1.We show that divisibility by m can be characterized through recursive reductions of the form N = ba + c → a ± Ec, where E satisfies bE ≡ ±1 (mod m).This yields a general osculation framework that extends classical decimal divisibilityrules to arbitrary numeral systems.Building on this foundation, the paper introduces the concept of a minimal oscillator, defined as an oscillator having minimum absolute value among all equivalentchoices. General existence results are established, together with bounds on oscillator magnitude and formal proofs of recursive divisibility preservation. The frameworkfurther leads to efficient algorithmic constructions based on the Extended EuclideanAlgorithm, providing logarithmic-time generation of divisibility rules for arbitrary admissible moduli.The results unify a wide variety of known divisibility tests within a single algebraicstructure and demonstrate that osculation is not a property of decimal arithmetic alonebut a general consequence of positional representation and modular invertibility. Thisperspective establishes a theoretical foundation for recursive divisibility testing acrossnumeral systems and suggests new directions for the study of digit-based reductionmethods.