A three-block higher-rank deformed Hermitian--Yang--Mills correspondence on ${\mathbb{P}}^1\times {\mathbb{P}}^1$

We study the invariant rank-four deformed Hermitian–Yang–Mills equation associated with the chain $\mathcal{O}(1)\to \mathcal{O}(2{)}^{\oplus 2}\to \mathcal{O}(3)$ over ${\mathbb{P}}^1$, with vertical weights $(4,2,0)$ and a fixed product Kähler class and phase. For arbitrary linear arrows $\psi =Bz$ and $\phi =(Az{)}^T$, smooth strictly calibrated reduced solutions exist exactly when $A$ is invertible and $z^T{A}^{T}Bz$ is not identically zero. The same condition characterizes stability, semistability and polystability in the specified equivariant coherent category. The classification includes all eleven nonempty coefficient strata and both coincident and distinct divisor points on the rank-one lower boundary. The analytic proof retains the ordered, noncommuting middle equation. After constructing positive-parameter solutions, it rules out unbounded metric values by combining the source budgets with moving spectral tests; on the rank-one lower boundary, the divisor mass must also be retained. Global value, gradient and symbol estimates then give a smooth strictly calibrated endpoint on the original chain. No equivalence with full coherent or derived-category stability, uniqueness, or estimate uniform across changing chains is asserted.