On the Pure Geometric Origin of the Fine-Structure Constant via Spontaneous Topological Doubling

The fine-structure constant, $\alpha$, remains one of the most profound unresolved fundamental constants in standard physics, conventionally treated as an empirical parameter of the electromagnetic coupling. In this Letter, we present a pure geometric derivation of $\alpha$, absent from any physical dimensions or empirical inputs. By modeling the vacuum state as a $T(1,2)$ toroidal manifold governed by the Law of Spontaneous Doubling (SD), we calculate the invariant measure of the manifold's volumetric projection. We demonstrate that the inverse fine-structure constant emerges strictly as a topological invariant: $\alpha_{geom}^{-1} = 2^7\pi/3 + 3 \approx 137.041$. This result suggests that the electromagnetic coupling strength is not an arbitrary parameter, but a deterministic geometric limit of a self-doubling topological space. Supplementary Material Included: The manuscript also provides a logical deduction extending the $T(1,2)$ manifold framework to the $2^4:(2^4-1)$ geometric asymmetry, thereby establishing a purely topological origin for the speed of light ($c_{geo}$), the Planck constant ($h_{geo}$), and gravitational entropy (the Big G-Formula). Theoretical Context: This work serves as an independent letter while inherently resting upon the mathematical foundations established in the Arc Geometry series (I, II, and III).