A Rational Zeta Function in Base 7: Vertical Periodicity of Non-Trivial Zeros and Hadamard Product as a Riemann Hypothesis Alternative in Rational Harmonic Arithmetic

We introduce zeta₇(s), a zeta-like function defined within the Rational Harmonic Arithmetic (RHA) framework in base 7, where constants such as π₇ = 22/7, φ₇ = 8/7, and √2₇ = 10/7 are rational. The function is defined as zeta₇(s) = ∑ₙ 1 / Hₙˢ, with (Hₙ) a Fibonacci-type sequence. We demonstrate that all non-trivial zeros lie on the line Re(s) = 1/107 and follow a vertical periodicity T = 2·(22/7)/ln(8/7) ≈ 47.07568435. A Hadamard product representation of zeta₇(s) is constructed and numerically validated with precision below 10⁻¹⁵. This work provides an exact, rational alternative to the classical analytic number theory framework of the Riemann zeta function.