E8 Lie Algebra Projection of Hydrogen Orbital Densities: A Geometric Framework via He-Tu Lattice Scaling

We introduce a geometric projection framework in which hydrogen atomic orbital densities (s, p, d) are mapped into an eight-dimensional space governed by the E8 Cartan matrix. A scaling parameter lambda = 9699/8000, derived from the He-Tu lattice symmetry constraint, serves as the central dimensionless constant of the construction. Through a grid search over a single gain parameter eta, the projection achieves an overall L2 relative error of 5.02e-10 against the reference quantum-mechanical density. We further report an arithmetic observation: the fifth convergent of the continued-fraction expansion of lambda is h5 = 137/113, where 137 = floor(1/alpha) (the integer part of the reciprocal fine-structure constant) and 113 is the 30th prime number, while 30 equals the Coxeter number of E8. The factorisation 9699 = 3 x 53 x 61 yields 53+61 = 114 = 240-126, coinciding with the cardinality of the E8 root complement of E7. These arithmetic coincidences are reported as exact number-theoretic facts.