The Dialectical Structure of Additive Recursion. From Integer Weights to Rational Extensions and Cyclic Order

Abstract

This work develops a complete mathematical theory of dialectical recursion on additive decompositions of a total mass N. Starting from an initial pair (L0, R0) with L0 + R0 = N, the recursion generates a deterministic sequence governed by a binary dialectical rule: doubling the smaller part or subtracting the larger part from twice the smaller. The theory establishes the exact structure of all resulting sequences, including preperiods, cycles, and termination conditions. A central invariant s_n = (L_n * a_n + R_n * b_n) / N is shown to remain constant for all n. The associated identity L_n * (s_n - a_n) = R_n * (b_n - s_n) expresses the classical law of the lever within the recursion. The cycle structure is determined by the order of 2 modulo the odd component of N, providing complete criteria for when the sequence terminates, enters a cycle, or exhibits a preperiod. The results extend from integer decompositions to rational decompositions (n1, n2 in Q+). By scaling with the least common multiple of denominators, the rational recursion becomes isomorphic to an integer recursion, preserving both the invariant and the lever law. This yields a unified framework for integer and rational additive systems. The work includes explicit examples, structural theorems (1–6), proofs, and the rational extension, forming a coherent foundation for further mathematical and philosophical development within the context of Transformation Loop Theory.