Coordinate Representations of the Natural Numbers Based on Parity

Description: This document develops the BBF framework as a static, index‑based representation of the natural numbers. The classical dyadic‑height decomposition

is interpreted geometrically as a two‑dimensional matrix in which the horizontal axis a encodes dyadic height and the vertical axis b encodes the odd core. As a structural counterpart to dyadic height, the notion of odd depth is introduced, providing a parallel coordinate that reflects the position of numbers within the odd‑indexed structure. The manuscript establishes the basic definitions, the geometric interpretation of the BBF_a and BBF_d representations, and the structural relations between height reduction and depth progression. These elements form the static foundation of the BBF framework.