CAT'S Theory Presents: No Other Way — Uniqueness of the Triadic Derivation of the Standard Model Constants

This paper proves that the CAT'S Theory derivation of the Standard Model coupling constants is the unique construction reproducing the observed values, resolving the numerology objection decisively. Three uniqueness results are established. First, the spectral constraint (d+D) + d² + 2ᵈ = 24 admits exactly one positive integer solution (d = 3, D = 4) with non-degenerate dimensional registers, and no modification of the hyperoperation orders, fiber dimension, or spectral budget produces a viable alternative. Second, exhaustive computational search over 27, 640 candidate formulas confirms that 36 match 1/α within 0. 5%, 4 also match sin²θW within 5%, and zero also match αₛ — total elimination by the third observable filter. Third, the Fibonacci–Lucas recursive architecture is the unique second-order linear recurrence compatible with the Pisano period π (9) = 24, the Ouroboros closure F (D) = d, and the minimal winding F (d) = 2, with the restriction to second-order recurrences forced by the Prism Identity excluding a fourth ontological axis within Σ = 26. A Structural Inevitability corollary proves formally that any framework satisfying the triadic persistence condition, the spectral constraint, and the Pisano–Ouroboros recurrence conditions must reduce to the CAT'S Theory system with all dimensionless observables determined and zero free parameters. The paper also provides an explicit mapping from the invariant R ≠ 0 to the elimination of alternatives, resolving the distinction between "condition" and "selection principle. " Presented in the Kuskov format (Problem → Result → Testable Prediction → Derivation → Philosophical Interpretation). Part of the CAT'S Theory corpus