The Algebraic Anatomy of the Canonical Assembly — Spectral, Representational and Functional Characterisation of Oef(G64)

This work presents a complete algebraic characterisation of the single assembly Oef(G64), the effective operator inherited from level LT1 of the TAGC programme, acting on the state space R^1024 = R^64 ⊗ R^16. We prove that Oef is diagonalisable over R, with spectrum μρ + ν, where μρ ∈ {0, 64|λ|, −20|λ|} and ν ∈ {0,1,2,3,4}, yielding exactly 15 spectral types with multiplicities dim(Vρ)·C(4,ν). Under the wreath-product symmetry group Sym(16×4) = S4 ≀ S16, the copy space decomposes uniquely as R^64 = V0 ⊕ V15 ⊕ V48, with dimensions 1, 15, 48, arising from the rank-3 association scheme {I, Aintra, Ainter} induced by the group's orbits on pairs of nodes. We show that the three sectors are irreducible representations of S4 ≀ S16, mutually non-isomorphic, and that every state of the assembly belongs to exactly one spectral type (ρ, ν). The operator Ainter is shown to distinguish the three sectors through a sign trichotomy (+, −, 0) that holds independently of the free coupling parameter |λ|. Finally, the parameter space is stratified into five regions delimited by five structural thresholds, exhibiting a duality Rk ↔ R5−k under Hamming complementation, with the fifth region lacking a mirror image. Together, these results establish five equivalent descriptions — group-theoretic, combinatorial, representational, spectral, and functional — of the same underlying algebraic structure, completing the algebraic closure of the single assembly within the scope of LT2, ahead of the geometric construction reserved for level L2 of the TAGC programme.