Mathematical Structures in Dark Geometry: A Rigorous Analysis of the 2/3 and 1/3 Constants

We present a rigorous mathematical analysis of the constants 2/3 and 1/3 that appear in both the Hardy Ramanujan partition formula and Ramanujan's mocktheta functions of order 3. These exact values coincide with cos^2 θH and sin^2 θH where θH = 35.26 is the Hertault angle of Dark Geometry. We trace the precise mathematical origin of these constants: the 2/3 arises from ζ(2) = π^2/6 and saddle-point analysis, while the 1/3 comes from the order classication of mocktheta functions. We identify a remarkable numerical coincidence: the equation 4/d! =(d − 1)/d is satisfied only for d = 3. Whether this coincidence reflects a deep connection between modular forms and three-dimensional geometry, or is merely numerical happenstance, remains an open question. Throughout, we distinguish clearly between proven mathematical facts and speculative interpretations.