Structural Completeness of the Torus-Knot Framework: The Standard Model and Gravity from T3 Topology

Abstract

We present the consolidated programme of the Unified Quantum Gravity Architectural Framework (UQGAF), in which fundamental particles are modelled as coprime torus knots T(P, Q) on a three-torus vacuum manifold T³ = S¹ × S¹ × S¹ embedded in S³ ≅ SU(2). From this single geometric postulate, together with the neutrino-complexity axiom n_ν = 5, the Geometric Impedance Formula m/m_e = Q²/(P · 2π²) reproduces approximately forty Standard Model observables: charged-lepton, quark, nucleon and gauge-boson masses; the fine-structure constant α⁻¹ = 137.036; the Weinberg angle sin²θ_W = 3/13; the strong coupling α_s = 484/4117; the CKM and PMNS mixing matrices; the CP-violating phases δ_CKM and δ_PMNS, with θ_QCD ≡ 0; the cosmological constant Λ across 122 orders of magnitude; and the galactic acceleration scale a₀ = (15/13)cH₀/(2π). No continuously adjustable constant is tuned to data; the framework's inputs, including the discrete catalogue of integer pairs, are enumerated explicitly rather than absorbed into the axiom count. The product category C_SM = SU(3)₂ × SU(2)₂ × Z₆ is established as the unique minimum of the Categorical Entanglement functional within the product-of-three-factors class subject to chirality, matter-content and anomaly-cancellation constraints. The cyclic fusion ring Z₅ is the unique structure yielding exactly three Dirac-compatible non-degenerate generations, and supplies a necessary selection rule forbidding isolated quarks and top-containing baryons, with colour-singlet-ness under SU(3)_C supplying sufficiency. Chirality emerges from the non-reciprocal R-matrix at Chern-Simons level k = 2. Because the adjoint of SU(3)₂ carries quantum dimension φ = (1+√5)/2, the Standard Model category is not pointed, and exact evaluation of its three-manifold quantum invariants is #P-hard. The continuum limit must therefore be computed algorithmically rather than in closed form: a proven property of the gauge structure selected by minimisation, standing in the relation quantum chromodynamics bears to its hadron spectrum. On substituting the Impedance Formula, the Dirac large-number relation reduces to a statement among three independently measured quantities, ln N = ln(α⁻¹)·ln[α⁻¹(m_p/m_e)(2π²)²] + ln(25/23), in which the exponent is the logarithmic separation between the Compton and Bohr scales of hydrogen. It predicts G = 6.672886 × 10⁻¹¹ m³kg⁻¹s⁻². The cube in the cosmological-constant expression is the holographic N-bound on the de Sitter horizon. Fourteen falsifiable predictions are scheduled for adjudication within the coming decade, among them δ_PMNS ≈ 225°, upper-octant sin²θ₂₃ = 49/90, m_β = 10.25 meV, r ≈ 3 × 10⁻⁴, a bifurcated step-resonance in the primordial gravitational-wave spectrum, and the value of G above, by DUNE, Hyper-Kamiokande, JUNO, KATRIN, Project 8, CMB-S4, LiteBIRD, LISA, Cosmic Explorer, and next-generation determinations of the Newtonian constant. The scalar spectral index identity 1 − n_s = α_s tan²θ_W = (P_W Q_d)/(P_{W³} P_u) = 1452/41170 matches Planck 2018 at 0.04σ and is derived through four equivalent routes. The hilltop scale μ² = 4P_{W³}P_u/(P_W Q_d) M_Pl² is fixed in closed form by the catalogue integers. This paper is the consolidating synthesis of five prior Zenodo papers and presents them as a single derivation chain.