FN02: Composition of Paired Planes - From Abelian Phase to Non-Abelian Unitary Symmetry

This paper studies the structural passage from an elementary two-dimensional paired sector to higher-dimensional unitary mixing. The primitive object is not a bare real plane, but a compatible two-dimensional structure P = (V2, g, J, ω), where g is a positive inner product, J2 = −Id is a compatible complex structure, and ω(u, v) = g(Ju, v) is the associated symplectic form. A single such plane is naturally a complex line and carries a U(1) phase symmetry. The direct sum of n independent paired planes carries the diagonal Abelian symmetry U(1)n. If the planes are structurally equivalent and coherent transformations between them are admitted, the symmetry enlarges to U(n).We then distinguish three operations that are often conflated: restriction to determinantone transformations, which gives SU(n); quotient by the common scalar phase, which gives PU(n) = U(n)/U(1); and removal of the central Lie-algebra direction, which gives u(n)/u(1) ∼= su(n). The relative-phase torus U(1)n/ΔU(1) is identified as the maximal torus of PU(n), while its finite central cover is the maximal torus of SU(n). Permutation symmetry supplies the Weyl group Sn, and continuous off-diagonal plane-mixing generators complete the Cartan–Weyl structure. The cases n = 1, 2, 3 yield respectively the structural patterns associated with U(1), SU(2)-type relative mixing, and SU(3)-type relative mixing. The paper does not claim that dimensional composition alone derives the physical gauge groups of the Standard Model. Its claim is more limited: non-Abelian unitary symmetry becomes structurally available when equivalent phase planes are assembled into one Hermitian carrier and continuous coherent mixing between them is admitted.