Chordal Geometry and Inverse Symmetrical Mapping in the Wave System

This paper introduces an advanced and expanded mathematical framework for "The Wave System," aimed at investigating the deterministic nature of composite number distributions and their associated prime gaps within the domain of odd integers (2M + 1). In this work, the paradigm shifts from observing discrete linear wave motions to formulating a comprehensive algebraic and geometric architecture designated as "Chordal Geometry," achieved by mapping the two-dimensional wave function W(n, k) onto a discrete Cartesian coordinate plane. The study provides rigorous algebraic proofs demonstrating absolute mirror symmetry across the central axis (45^∘), and uncovers a strict structural classification of chords based on the nature of the constant c. Specifically, the system bifurcates into sovereign anchor chords (even c) that pass through solid energy nodes and generate perfect squares of odd numbers, and intermediate symmetric chords (odd c) that govern the commutative spaces between waves. Culminating this geometric discipline, the paper proposes an inverse system governed by precise kinematic constraints (kₘₐₓ), which enables the direct and deterministic detection and classification of numerical coordinates and absolute gaps. This approach opens a new horizon for addressing the prime distribution problem computationally and theoretically, bypassing traditional probabilistic methods.