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

Version 2 Update (March 12, 2026): This updated version replaces the manuscript file to formally include the official Zenodo DOI link, the correspondence email, and the author's ORCID in the document's footnote for proper citation and open-science compliance. The core theoretical content, geometric derivations, and the Big G-Formula remain strictly unchanged from the initial release. The fine-structure constant, , 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, 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: ₆₄₎₌^-1 = 2⁷/3 + 3 137. 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 2⁴: (2⁴-1) geometric asymmetry, thereby establishing a purely topological origin for the speed of light (c₆₄₎), the Planck constant (h₆₄₎), 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).