Anomaly Rigidity and Borel–Weil–Bott Obstructions to Matrix-Quiver Realizations in Heterotic Full-Flag Cosets

We study a compatibility problem for the integrable full flag F₃ = SU(3)/T² in perturbative heterotic string theory. Combining the heterotic $(0,2)$ coset anomaly equation, the integral A₂ cocharacter lattice, Borel–Weil–Bott cohomology, and D₁₆ centralizer multiplicities, we classify integral left-torus embeddings at geometric-point levels k = 3p². Positive semidefiniteness leaves only the index-one branch and the one-sided branch of index p². In the index-one multiplicity problem, the anomaly equations admit exactly 192 ordered charge assignments. Two one-dimensional H¹ channels occur only in the zero-sum class; activating both complementary simple-root extensions reconstructs the trivial rank-three bundle, while the ordinary two-step line-bundle Yoneda endpoint vanishes by Borel–Weil–Bott singularity. At the first nontrivial one-sided branch $p = 2$, an exhaustive exact-integer census of the sixteen complex Spin(32) gauge fermions yields the nested count 206 42 0, showing that, within the stated elementary torus-selected U(3)⁴ ansatz, the anomaly, centralizer-multiplicity, and complementary BWB simple-root requirements cannot be satisfied simultaneously. The obstruction is deliberately scoped: it is not a no-go theorem for heterotic ultraviolet completions in general and does not exclude more general Narain data, different gauge parents, additional current-algebra sectors, nonperturbative constructions, or composite/emergent matrix nodes. A deterministic Python verifier for the finite classification is supplied with the record.