A Rational Fibonacci-Like Sequence in Base 7 Converging to Phi₇ = 8⁄7 — An Exact Golden Ratio Analogue

This work introduces a rational Fibonacci-like sequence defined in base 7, converging to the exact constant Phi₇ = 8⁄7 — a base-specific analogue of the golden ratio, but fully rational and structurally exact. Each term follows the recurrence Hₙ = Hₙ₋₁ + (1⁄7)·Hₙ₋₂, with initial values H₀ = 7 and H₁ = 8. The sequence generates exact fractions of the form k⁄7ⁿ, aligned with base-7 numeration and prime-length jumps. Unlike the classical Fibonacci sequence, which converges to an irrational limit, this one produces predictable, harmonic spirals from rational arithmetic alone. The limit Phi₇ solves the minimal equation x² − x − 1⁄7 = 0 and generalizes to all integer bases via Phi_b = (b+1)⁄b. This construction opens new perspectives in numerical geometry, cyclic computation, and non-decimal mathematical frameworks.