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

This work introduces a Fibonacci-like sequence En defined in a base-7 framework called Rational Harmonic Arithmetic (RHA). The sequence converges empirically to the rational constant e₇ = 19/7 ≈ 2.7142857143, offering a structured and finite analogue to the transcendental Euler constant e ≈ 2.718. Each term is a rational fraction k/7ⁿ, with patterns aligned to base-7 prime length jumps. This construction extends the RHA model previously applied to π and φ, and opens new perspectives for modeling exponential growth in discrete, rational arithmetic systems. Full documentation includes derivation, examples, graphical visualizations, and generalizations to √2