A Rational Geometric Framework in Base 7: Redefining π and the Golden Ratio

This study introduces a rational geometric system grounded in base 7 arithmetic, where the classical constants Pi and the Golden Ratio are redefined as exact rational values: Pi₇ = 22/7 and Phi₇ = 8/7. A full circle is set to 121 (base 7) = 64 (base 10) angular units, enabling integer divisions and cyclic fractions of the form k/7. The system reveals strong structural harmony with prime numbers such as 53 and 347. Unlike traditional transcendental formulations, these constants emerge naturally from the base 7 framework, offering a compact and coherent alternative. Applications include geometric constructions, proportional spirals, cryptographic periodicity, and reinterpretations of space under rational base assumptions. This work challenges the universality of classical constants and proposes a modular arithmetic model with deep mathematical implications.