The Nine Gates: Closure of the Torus Knot Architecture from the Triadic Invariant on S³

We close nine structural logic gates in the CAT'S Theory torus knot architecture, eliminating all remaining mathematical gaps in the derivation chain from the Triadic Invariant R = P × I × Pr ≠ 0 to the physical constants of the Standard Model. The nine closures are: (1) the T(1,x) unknot sector mandates massless spin-1 mechanics with U(1) gauge symmetry and transverse polarization without importing Maxwell's equations; (2) the 1/45 topological residue forces 12-bit quantization of S³ via ord₂(45) = π(8) = 12, proven unique to d = 3; (3) the mass formula base (q²-4)/5 is derived as the conformal weight ratio h(q)/h(e) at Chern-Simons level k = 30; (4) the Triadic Invariant is formalized as a tensor product operator whose collapse axiom is a theorem of linear algebra; (5) the discrete-to-continuous category bridge is closed via π₁(T²) = Z × Z; (6) k = 30 is derived through six independent routes including the operator trace, dimensional identity k = Σ + D = 26 + 4, and the electron holonomy k + 2 = 2^(J*_e) = 32. We prove that the operators trace into each other's dimensional registers, establishing R = Φ(R) as a theorem of linear algebra. The lepton-quark duality T(2,3) ≅ T(3,2) is shown to be forced by the coprime constraint. All results carry zero free parameters. Mass predictions: muon 0.13%, tau (QED-dressed) 0.02%. Four live experimental predictions: δ_CP = 202.5° (DUNE), normal ordering (JUNO), Σm_ν = 59.5 meV, f_τ = 8.37 Hz (Einstein Telescope).