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 $F_3 = SU(3)/T^2$ in perturbative heterotic string theory. Combining the heterotic $(0,2)$ coset anomaly equation, the integral $A_2$ cocharacter lattice, Borel–Weil–Bott cohomology, and $D_{16}$ centralizer multiplicities, we classify integral left-torus embeddings at geometric-point levels $k = 3p^2$. Positive semidefiniteness leaves only the index-one branch and the one-sided branch of index $p^2$. In the index-one multiplicity problem, the anomaly equations admit exactly 192 ordered charge assignments. Two one-dimensional $H^1$ 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 $\mathrm{Spin}(32)$ gauge fermions yields the nested count $$206 \longrightarrow 42 \longrightarrow 0,$$ 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.