The Standard Model from a Torus Knot

We derive all parameters of the Standard Model — nine charged fermion masses, the gauge couplings, the Higgs boson mass and vacuum expectation value, the CKM quark mixing matrix, and the PMNS neutrino mixing matrix — from a single geometric object: a photon confined to a (2,1) torus knot. This framework, which we name “null worldtube” (NWT), requires one measured mass (m_e) and three topological integers (p = 2, q = 1, k =R/r = 3). The fine-structure constant α follows from Skilton’s integer right triangle (88,105,137), whose generators (11,4) are NWT quantum numbers. The muon mass follows from the Koide angle θK = (6π +2)/9 with topology-determined parameters. The tube energy scale Λ_tube follows from the Lenz formula m_p/m_e = 6π5 = 2k ×πp2+q2. The electroweak sector follows from torus mode counting: sin2 θW = 3/13, the Higgs mass from the breathing mode mH = (1/2−α)Λ, and the vacuum expectation value v =(1−5α)Λ. The CKM phase is the knot deficit angle δCP = π−2. The PMNS solar angle shares the Weinberg denominator: sin2 θ12 = 4/13. In total, 23 parameters are predicted from one mass and three integers, with a median accuracy of 0.7%. The Standard Model is geometry.