A Rational Euler Constant in Base 7: Fibonacci-Type Convergence in Rational Harmonic Arithmetic

This work introduces a Fibonacci‑like sequence Eₙ defined by the recurrenceEₙ = Eₙ₋₁ + (1/7) Eₙ₋₂within a base‑7 framework called Rational Harmonic Arithmetic (RHA). Empirically, the ratio Eₙ₊₁∕Eₙ converges to the algebraic constant r ≈ 1.12678, providing a purely rational model of exponential growth. Each term Eₙ is a rational number whose denominator is 7ⁿ and whose numerator increases in jumps at base‑7 digit‑length thresholds tied to primes. Extending prior RHA rationalizations of π and φ, this construction offers finite, discrete analogues to classical constants. Full documentation includes detailed derivations, worked examples, graphical plots of Eₙ, and a generalization to approximate √2.