A Rank-Fifteen Residual-Shell Algebra, Cubic Charge Algebras for Z₃ Quotients, and Canonical Embeddings into the Ogg Prime and Genus-Zero Level Algebras

A finite semisimple algebra is derived from two residual Erdős–Straus structures: the five-lock affine factor and the six classical residual square classes modulo 840. Over the coefficient field 𝕂: = ℂ (T₅₀), these data determine the rank-30 coordinate algebra𝒪 (𝔽₅ × C₆). Translation by the unique order-two element of C₆ yields the rank-15 invariant algebra𝒪 (𝔽₅ × C₃) ≅ 𝒪 (𝔽₅) ⊗ 𝒪 (C₃), and this algebra is identified with the cubic charge-sector algebras for SL₆/μ₃ and (SL₆ × SL₃) /μ₃. The zero-sheet separation theorem produces a support-completed neutral idempotent and hence a rank-16 refinement compatible with the regular sixteen-sheet cover of the odd n-line theory. In support-completed form the four primitive sectors split into a fixed pair and an exchanged pair governed by an explicit mirror involution matching the reversal symmetry on the Apollonian opposite-edge classes. On the Ogg side this yields the canonical 1+7+7 decomposition of the prime algebra and the canonical 1+1+7+7 decomposition of the enlarged genus-zero level algebra obtained by adjoining level 1; the adjoined level is identified with the fixed isotropy/orientation line complementary to the isolated zero line. The enlarged Ogg algebra also determines a canonical ordered primitive idempotent system, which transports to the residual-shell and affine four-sector algebras and induces a canonical ordered primitive basis on the rank-16 deck algebra. We further derive an exact odd-unit embedding of the fourteen odd Ogg levels into 𝔽₅ × C₆ and recast the quadratic C₆/C₃ reduction in explicit character-theoretic form as even-character projection. Finally, the same fixed line is identified locally as the charge-zero cubic sector, the trivial real line in the real C₃ ⋊ C₂ form, the h-axis in Koide coordinates, the reversal-fixed opposite-pair class in the Apollonian quotient, and the charge-zero linear model Λ³V₆, while the exchanged pair is realized by the nontrivial real block, the orthogonal Koide plane, the two non-fixed Apollonian classes, and the bifundamental and conjugate linear models.