From SU(5)×SU(5) Block Decompositions to an Extended Pati--Salam Structure: $E_8$ and $G_{422111}$ theory

Taking $SU(5)_A\times SU(5)_B$ as its starting point, this paper addresses a simple structural question: can a single choice of block decompositions explain both the Pati--Salam non-Abelian sector and the origin of three additional Abelian directions? The first five-dimensional space is decomposed as $4+1$, and the second as $2+2+1$. Special unitary transformations within the blocks yield $SU(4)_C\times SU(2)_L\times SU(2)_R$, while the block phases, subject to the unit-determinant constraints, leave one and two independent directions, respectively. Together they give the local group structure of $G_{422111}$. The construction thus provides a common block interpretation of the number, generators, and left--right exchange properties of the three $U(1)$ factors. Using the known embedding of an $SU(5)^2$-type subgroup in $E_8$, we obtain a Pati--Salam embedding and adjoint representation content distinct from those of the standard route through $SO(10)$. The representation analysis imposes two restrictions: the selection of an extra anomaly-free direction depends on the chiral spectrum and a left--right evenness condition, and ordinary Higgs breaking using only Standard Model singlets in the adjoint necessarily preserves an additional continuous $U(1)$. We treat the block construction as a structural basis for a unification scheme, with spectrum selection and vacuum dynamics requiring further physical input. The main text emphasizes the construction and its physical interpretation; complete branching rules and proofs are collected in the appendices.