TWISTED CLIFFORD FUNCTIONAL QUANTIZATION: HEIGHT SPECTRA, ROOT SYSTEMS, AND THE UNIVERSAL OPERATOR

We establish that the minimal TCFQ operator associated to an irreducible simply-laced root system Φ is determined solely by the Coxeter number h of Φ. Specifically, 𝒜_Φ(𝑡) = 𝑡² ∏_{𝑘=1}^{𝘩−1} (𝑡² + 𝑘²)², deg 𝒜_Φ = 4(ℎ − 1) + 2. The proof rests on three pillars: (i) an exact closed form for the generating function 𝐺_𝑚(𝑡) (Block R86/R87); (ii) the explicit spectral operator for type 𝐷_𝑛 and its numerical verification (Block R87); and (iii) a type-independent argument showing that the choice 𝜃 = 𝑥𝜌 collapses the geometry of Φ⁺ to the scalar datum of root height, combined with the classical fact that Ht(Φ) = {1, … , ℎ − 1} for every simply-laced irreducible system (Blocks R88–R89).