The Geometric Nature of Photoelectric Effect and Superconductivity: Electron and Quark as a Topological Torus in Noncommutative Spectral Geometry

We present a unified geometric theory of the photoelectric effect and superconductivity, based on the identification of fundamental fermions as topological tori T 2 in noncommutative spectral geometry. The electron’s quantum states correspondto modes (m, kz) on this torus, determined by boundary conditions imposed by the external geometric landscape (atomic potential, crystal lattice). The photoelectric effect is reinterpreted as a resonant topological transition: a photon (spherical torsion wave) excites the electron-torus, which undergoes a sequence of N fundamental topological steps, each of duration τ0 = Rc/c, before escaping into the continuum.The transition time ∆t = N ·τ0 naturally explains recent attosecond measurements showing material-dependent emission times (26 as in Cu, 140-175 as in layered crystals, 200 as in chain structures).Superconductivity emerges as topological pairing: below a critical temperature Tc, two electron-tori resonate through the lattice landscape and undergo a topological fusion T 2#T 2 → Σ2 (genus-2 surface), forming a composite structure with effective bosonic statistics (J2 = +1). This explains the disappearance of resistance and the Meissner effect as topological incompatibility with external torsion fields. The critical temperature is derived as kBTc = ℏ/(Npairτ0), where Npair is the number of steps required for pairing.We further extend the framework to all fundamental fermions, showing that quarks are tori with non-trivial SU(3) color bundles classified by the first Chern class c1 ∈ Z3, and nucleons are connected sums of three tori Σ3 whose topological protection explains confinement and proton stability. This unified picture reveals that all fermions are axial vortices at characteristic scales ϕeff0 , with masses and sizesrelated by the vortex scaling laws Rc ∼ 1/(gϕ0) and Mv ∼ ϕ0/g2.This work completes the geometric unification program by demonstrating that the same principles governing astrophysical jets and black hole engines also underlie all fundamental particles and their interactions. The electron-torus is identified as an axial vortex at the electroweak scale (ϕ0 = 246 GeV, Rc ∼ 10−18 m), quarks as colored tori at scales ϕ0 ∼ 103−108 GeV, revealing the deep unity of physics across 40 orders of magnitude.