The Universal 2/9 Trace Coefficient as a Discrete Atiyah–Singer Index: A Topological-Index Unification of Six Independent Appearances of δ= 2/9 on the Canonical Truncated Cubic Honeycomb

The rational coefficient δ = 2/9 appears as a fundamental structural invariant across six independent sectors of the canonical Holographic Circlette framework: (i) the chiral Q3 anomaly trace Q3 L =−2/9 (Part 7, ANCHOR §2.11); (ii) the bare Weinberg mixing angle sin2 θW = 2/9 from the EW 9 = 7 + 2 partition (Parts 6/19, ANCHOR §6.6); (iii) the charged-lepton Koide circulant phase δ= 2/9 from the defect-to-plaquette ratio (Parts 5/18, ANCHOR §5.8); (iv) the νR Berry phase δ = 2/9 from the canonical walk-operator amplitude (Part 6); (v) the neutrino Koide phase δν = 3/9 = 2/9 + 1/9 extending the systematic defect-fraction structure (Part 5); and (vi) the CKM Wolfenstein parameter λ= 2/9 from the LQ= 1 quark subspace at walk-operator depth k= 2 (Part 4, ANCHOR §6.7). We demonstrate that these six independent appearances are not coincidences but six trace projections of a single discrete topological index: the Chern number c1 = d/N = 2/9 of the F2 bundle over the canonical 9-qubit plaquette (the local 4.8.8 vertex figure of the TCH substrate), with d= 2 defect nodes (R1 violation support) and N = 9 plaquette sites. The unification mechanism is the discrete lattice Atiyah–Singer index theorem of Hasenfratz– Laliena–Niedermayer (1998) and Lüscher (1998) applied to the canonical Ginsparg–Wilson Dirac operator DTCH (ANCHOR §15 item 93 closure). The universal 2/9 is the topological index of the canonical walk operator W, and every observable derived from a trace of W(or of a composite operator built from W) is quantised in units of this Chern number. This closes ANCHOR §15 item 86 at Locked tier through the discrete Atiyah–Singer index theorem.