PDSM Geometrization of the Fine-Structure Constant: Six- Transcendental Basis Fitting and CY₃ Topological Correspondence

AbstractThe fine-structure constant α ≈1 /137 is a dimensionless fundamental constantcharacterizing the strength of electromagnetic coupling in quantum electrodynamics. Itsvalue cannot be derived from first principles and was called “one of the greatest damnmysteries of physics” by Feynman [9]. Within the PDSM (Primitive Vortex Number Theory–Wormhole Spaceology) framework, this paper systematically investigates the fitting of α −1via integer-coefficient linear combinations of transcendental number bases. We first provethat the pure-π cubic polynomial 4 π 3+ π2 +π achieves 2.22 ppm precision and constitutesan isolated optimal solution within this constrained space. We then introduce the six-transcendental basis Ω6 ={π , e , ϕ , ln2 , ln 3 , √2 } and prove, via the Dimension ExpansionTheorem, that mixed-product structures inflate the fitting space from single digits tomillions of dimensions. Under the all-positive-coefficient constraint, a 4-term productformula reaches 4.19 ppb, and a 5-term product formula reaches 0.43 ppb—93 times moreprecise than the negative-coefficient cancellation formula. We further construct the PDSM-RH holomorphic map Fshui: Xshui → P1, geometrizing α as the ratio of period integrals of theShui stress tensor over a Calabi-Yau threefold. Finally, we establish a correspondence between the topological invariants of the quintic hypersurface and the fitting parameters:, providing a novel theoretical pathway for deriving αfrom geometric first principles