The Unified Origin of Mathematical Constants: From Holographic Normalization to MockTheta Functions

We demonstrate that the mathematical constants 2/3 and 1/3 appearing in Hardy-Ramanujan's partition formula, Ramanujan's mock theta functions, and the modular group relation (ST)^3 = −I all originate from the same geometric fact: the universe has d = 3 spatial dimensions. The key insight is that these are not independent coincidences but manifestations of a single principle: any quantity normalized to unity at a holographic boundary separates into fractions (d − 1)/d and 1/d when projected to surface and bulk components. For d = 3, this gives 2/3 and 1/3. We establish that d = 3 is the unique dimension maximizing holographic degrees of freedom in quantum gravity, and show that the group SL(2, R) whose discrete subgroup SL(2, Z) governs modular forms is precisely the isometry group of AdS3. Thus, the 3 in (ST)^3 = −I is not a numerical coincidence but an identity with d = 3.