Constrained SO(3,3) Spacetime: Part V. Torus Knot Topologies and the Emergence of Lepton Mass Hierarchy

Abstract

The origin of the exponential mass hierarchy among the three charged lepton generations (electron, muon, tau) remains one of the prominent open questions in fundamental particle physics, typically accommodated within the Standard Model via fine-tuned Yukawa couplings. Building upon the SO(3,3) multi-time spacetime framework introduced in Parts I–IV, this paper explores an exploratory, parameter-free geometric ansatz to address this hierarchy. We model charged leptons as closed topological solitons confined within an effective 1+2 dimensional time manifold (\mathcal{T}^3). Under the assumption that the identical quantum numbers of the lepton generations reflect a common topological family, we map the three generations to sequential torus knots: the unknot (0_1) for the electron, the trefoil knot (3_1) for the muon, and the (2,5)-torus knot (5_1) for the tau. Using the mathematical invariant of ideal ropelength as a heuristic proxy for geometric tension against the macroscopic meta-time flow, we define a dimensionless effective action (\mathcal{A}). Within this temporal manifold, a formal Wick rotation maps this linear topological invariant into an exponential mass scaling factor, M \propto \exp(\mathcal{A}). This simple geometric ansatz yields bare mass ratios of M_\mu/M_e \approx 183.3 and M_\tau/M_e \approx 3626, displaying an intriguing proximity to the observed physical pole mass ratios (\approx 206.7 and \approx 3477). While these results omit rigorous high-order radiative corrections and non-perturbative gauge interactions, the qualitative and numerical alignment suggests that multi-time geometry may provide a compelling avenue for understanding mass hierarchies. We present this model as a theoretical hypothesis inviting further rigorous field-theoretic and mathematical scrutiny.