Mathematical Foundations of the Torus Knot Architecture: Derivation of the Dimensional Registers, Hopf-Projected Conformal Weight, Dynamical Sector, and Uniqueness Criteria from the Triadic Invariant on S³

The torus knot architecture of CAT'S Theory derives Standard Model constants from T(p,q) knots on S³ via the Hopf fibration with zero free parameters. This paper provides nine mathematical derivations that the architecture presupposes but has not previously consolidated in a single document: (1) the dimensional registers dΨ=7, dP=8, dI=9 are uniquely forced by d=3 through extensive, relational, and orientational counting (vector addition, bilateral adjoint tensor products on SU(2)×SU(2), and the Clifford algebra Cl(3)); (2) the Hopf-projected conformal weight h=J×J*/(k+2) is proven unique among bilinear forms satisfying Casimir reduction and base↔fiber symmetry — the standard WZW Casimir predicts zero CP violation, while the Hopf-projected weight predicts δ_CP=202.5°, matching PDG at 0.2σ; (3) the dynamical sector is the Euler–Heisenberg effective Lagrangian evaluated on field configurations with non-trivial Hopf index — the topological sector of the Standard Model's own action; (4) the four uniqueness criteria governing the T(3,8) exhaustive search are individually required by the axioms of R=P×I×Pr≠0 on S³; (5) spin-statistics is derived as a theorem from S³ topology plus causal structure — the wrong-statistics assignment produces an explicit microcausality violation; (6) the fine-structure expansion obeys a triadic perturbation rule (Pattern→Intent→Presence at successive loop orders), anchored by the structural identity k−2=T(dΨ)=28, yielding α⁻¹=137.036004; (7) the Standard Model gauge group SU(3)×SU(2)×U(1) is the unique group compatible with the triadic decomposition, derived via the braid group B₃ and Burau representation; (8) coupling constant ratios are determined by the registers, with α_s/α=D²=16; (9) UV finiteness follows from the topological cutoff — modes that cannot close as torus knots are forbidden, making loop integrals finite sums. No predictions are altered. All kill sentences remain active. Companion to The Nine Gates, The Same Equation at Every Scale, Gravity as Dielectrophoresis, and The Fortress Holds.