Number-Theoretic Properties of the Rational λ = 9699/8000 and Observations on E8 Root System Geometry

We document the arithmetic structure of the rational number λ = 9699/8000 = 1. 212375 (exactly). Its continued fraction expansion is 1; 4, 1, 2, 2, 3, 5, 6, 2, terminating in 9 partial quotients. The penultimate convergent is h5 = 137/113, which approximates λ with exact error 13/904, 000 ≈ 1. 44×10⁻⁵. The prime factorizations are 9699 = 3×53×61 and 8000 = 2⁶×5³. We further document the angle distribution obtained by projecting the E8 root system (240 vectors in R⁸) onto R³ via the algebraically determined E8 ⊃ H4 decomposition, finding that the projection is governed by the golden ratio φ = (1+√5) /2 with no preferred angle near 104. 45°. All results are exactly reproducible; verification code is provided in the Appendix.