α=κ/τ: Paradigm Shift of the Fine‑Structure Constant from a Physical Coupling to a Geometric Constant

Abstract This paper presents a geometric interpretation of the fine‑structure constant α, relating it to the curvature κ and torsion τ of a photon’s helical trajectory, and resolves the Dirac–Feynman conjecture. The fine‑structure constant α, a dimensionless coupling constant of electromagnetic interaction, has long been a central puzzle in theoretical physics. Decades ago, pioneers including Dirac and Feynman hypothesized that α must have geometric properties linked to π, yet they did not uncover its intrinsic quantitative relationship. Drawing on the curvature–torsion theory of spatial curves in differential geometry, this paper proposes and proves the core relationship , revealing for the first time the geometric nature of α: α is not merely an empirical coupling constant, but an intrinsic geometric constant inherent in microscopic particle trajectories, establishing a cross‑dimensional isomorphic twin relationship with π. This result not only explains the small drift of α across quantum energy levels but also achieves a paradigm shift: from a physical coupling constant to a fundamental geometric constant, opening a new avenue for bridging quantum physics and differential geometry. This paper is a twin follow‑up to the original work published on May 11, 2026, titled : The Geometric Origin of the Fine‑Structure Constant. Retaining the core theory, the present study deepens and rigorously completes the research. The original paper first proposed and secured academic priority for the geometric origin of α tied to 3D helical photon motion; the present work, based on intrinsic geometric invariants, eliminates extrinsic parameters and formulates via curvature κ and torsion τ. The derivation is systematic, rigorous, and fully validated against authoritative experimental data. High‑precision measurements confirm the geometric origin of α, resolving the century‑old Dirac–Feynman conjecture. π is the fundamental constant of 2D planar geometry; α is the fundamental constant of spatial curve geometry. They are dimensionally corresponding, definitionally homologous, and structurally isomorphic. Both are dimensionless natural constants independent of empirical physical parameters. This overturns the conventional view, unifies macroscopic and microscopic geometric laws, and provides a new foundation for differential geometry. Note that, given current experimental precision, the conclusions herein require further testing, validation, refinement, and extension. As my physics teacher and father taught me, physics is an empirical science. No elegant theory becomes established law without rigorous experimental scrutiny. This paper adopts a prudent stance, awaiting future progress.