Structural Completeness of the Torus-Knot Framework: A Zero-Parameter Derivation of the Standard Model from T³ Topology

We present the consolidated programme of the Unified Quantum Gravity Architectural Framework (UQGAF), in which fundamental particles are modelled as torus knots T (P, Q) on a three-torus vacuum manifold T³ = S¹ × S¹ × S¹ embedded in the Lie group S³ ≅ SU (2). From this single geometric postulate, together with the neutrino-complexity axiom n_ν = 5, the Geometric Impedance Formula m/mₑ = Q²/ (P · 2π²) reproduces approximately forty Standard Model observables, including charged-lepton, quark, nucleon, and gauge-boson masses; the fine-structure constant α⁻¹ = 137. 036; the Weinberg angle sin²θW = 3/13; the strong coupling αₛ = 484/4117; the CKM and PMNS mixing matrices; the CP-violating phases δCKM, δPMNS, and θQCD ≡ 0 (mod 2π) ; the cosmological constant Λ spanning 122 orders of magnitude from the Planck scale; and the galactic acceleration scale a₀ = (15/13) cH₀/ (2π), with no continuously adjustable free parameters. The product category CSM = SU (3) ₂ × SU (2) ₂ × Z₆ is established as the unique minimum of the Categorical Entanglement functional within the physically motivated product-of-three-factors class subject to chirality, matter-content, and anomaly-cancellation constraints. The cyclic fusion ring Z₅ is established as the unique structure yielding exactly three Dirac-compatible non-degenerate fermion generations. Colour confinement is derived as a theorem rather than postulated, and chirality emerges from the non-reciprocal R-matrix at Chern-Simons level k = 2. The programme generates specific falsifiable predictions, including δPMNS ≈ 225°, upper-octant sin²θ₂3 = 49/90, absolute neutrino masses m_β = 10. 25 meV, tensor-to-scalar ratio r ~ 3 × 10⁻⁴, and a bifurcated step-resonance in the primordial gravitational-wave spectrum from categorical phase transitions, all scheduled for experimental adjudication by DUNE, Hyper-Kamiokande, JUNO, KATRIN, Project 8, CMB-S4, LiteBIRD, LISA, and Cosmic Explorer within the coming decade. The scalar spectral index identity 1 − nₛ = αₛ tan²θW = (PW Qd) / (Pₖ℃ Pᵤ) = 1452/41170 matches Planck 2018 data at 0. 04σ and is derived structurally through four equivalent routes: Cartan replacement on T³, the charged-current/neutral-current amplitude ratio, the Vertex Amplitude Principle extending Appendix D of the framework papers from particles to vertex configurations, and the unique Casimir factorisation CF (SU (3) ) · CF (SU (2) ) = 1 structurally selected by the Standard Model gauge group. The hilltop scale μ² = 4 Pₖ℃ Pᵤ/ (PW Qd) MPl² is fixed in closed form by the catalogue integers, recovering r ~ 3 × 10⁻⁴ as a derived rather than fitted prediction. This paper is the consolidating synthesis of five prior Zenodo papers (the main UQGAF paper, Coprime Resonance, Quark Flavour, Standard Model, and Closure) and presents them as a single derivation chain.