The BBF Quadrant System: Structural Properties of Integer and Rational Encodings

This work introduces the BBF Quadrant System, a symbolic number‑space architecture that maps natural numbers into a structured quadrant representation. The framework identifies persistent cells, emergent arithmetic patterns, and topological relations arising from the BBF formulation. By extending the underlying COL‑based domain structure, the system reveals stable symbolic configurations and numeric behaviors that support a generalized view of arithmetic genesis. The manuscript provides the formal definitions, structural rules, and verification steps used to derive the quadrant system, including EXCEL‑based validation and symbolic consistency checks. Additional Description (Methods) The study develops the BBF Quadrant System through a combination of symbolic derivation, domain‑based COL‑extensions, and computational verification. Structural properties are analyzed using persistent‑cell tracking, sign‑based parameter variation, and numeric topology inspection. All constructions are validated through reproducible EXCEL procedures and stepwise symbolic transformations, ensuring transparency and traceability of the resulting number‑space architecture.