Rational Harmonic Arithmetic and Trigonometry A Formal System for Object Classification and Trigonometric Operations in RHA

This document presents a complete reformulation of trigonometry and numerical structure using Rational Harmonic Arithmetic (RHA), a system built on rational units and geometric alignment in base 7. Unlike classical trigonometry, which depends on transcendental approximations and floating-point computation, RHA defines all constants, functions, and operators as exact rational expressions. The core functions sin_b, cos_b, and tan_b are reconstructed from first principles using harmonic polygonal approximations, with closed-form rational formulas valid for all integer bases b ≥ 2. A new identity replaces the classical Pythagorean relation, and all differential operations are redefined as discrete, rational steps. The document includes formal theorems, rigorous proofs, residual error analysis, and empirical comparisons between classical and harmonic formulations. Applications include exact geometric computation, stable symbolic arithmetic, and physical constant alignment on a rational spectrum. This work introduces a self-consistent rational alternative to classical trigonometry and supports further extensions in mathematical physics and symbolic computation.