Towards a Substrate-Level Glueball Spectrum: A Programme Note on Closed Cycles on L(Z3) in the TCH Framework

The Holographic Circlette (TCH) framework derives the bare ρ(770) mass from the line- graph spectrum of a meson flux tube and the nucleon mass from a closed cycle on the Q3 matter octagon, in both cases by zero-parameter spectral calculations on the discrete substrate. We extend this machinery to the glueball sector by computing closed-cycle spectra on L(Z3), the line graph of the simple-cubic gauge web. The substrate-first-principle that selects the combination rule between disjoint-channel quadrature (used for the ρ) and shared- volume linear superposition (used here) is the HDR Exemption Corollary of the Holographic Dimensional Reduction Theorem (ANCHOR §15 item 77): HDR’s 2D-boundary projection of massive defects requires a magnetic-monopole-like defect, which glueballs lack. Pure- gauge cycles therefore remain 3D-bulk volumetric excitations and combine linearly. The lightest 0++ scalar glueball arises from the totally symmetric (A1g ) sum of three orthogonal C4 plaquettes: mbare 0++ = 6ΛQCD ≈1992 MeV, dressing∼14% downward via attractive scalar-meson continuum mixing to the LQCD pole∼1710 MeV. The lightest 2++ tensor glueball cannot be built from plaquettes alone (the three-plaquette space decomposes as A1g ⊕Eg , with no T2g component), and requires the four non-planar Petrie C6 hexagons that decompose as A1g ⊕T2g under Oh: mbare 2++ = 8ΛQCD ≈2656 MeV. The mass ratio m2++ /m0++ = 4/3 ≈1.333 is parameter-free and matches the LQCD empirical ratio 1.397 within 5%. The HDR Exemption resolves the √2 vs √3 ansatz dispute of ANCHOR §15 item 104 for the glueball sector. The remaining open work is the first-principles derivation of the scalar and tensor dressing fractions from substrate-level self-energy mixing