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 derives the four mathematical foundations the architecture presupposes but has not previously derived in a single published 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 the natural weight for knot states carrying independent base and fiber quantum numbers — the standard WZW Casimir j(j+1)/(k+2) 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, each mapping to a specific factor of the Triadic Invariant. 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.