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

This preprint derives a finite semisimple algebra from two residual Erdős–Straus structures: the five-lock affine factor and the six classical residual square classes modulo 840. From these data it derives a rank-30 coordinate algebra on F₅ × C₆, proves that the involution by translation through the unique order-two element of C₆ yields a rank-15 invariant algebra on F₅ × C₃, and identifies that algebra with explicit cubic charge-sector algebras for SL₆/μ₃ and (SL₆ × SL₃)/μ₃. The paper also derives the support-completed neutral idempotent from the zero-sheet separation theorem, yielding a rank-16 refinement compatible with the regular sixteen-sheet cover on the residual Erdős–Straus side. In this 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, the prime algebra carries the canonical 1+7+7 decomposition, while the enlarged genus-zero level algebra obtained by adjoining level 1 carries a canonical 1+1+7+7 decomposition. The adjoined level 1 is identified explicitly with the fixed isotropy/orientation line, complementary to the isolated zero line, and the same fixed-pair/exchanged-pair pattern is proved to persist across the support-completed cubic algebra, the affine support algebra, the enlarged Ogg level algebra, and the regular sixteen-sheet deck algebra. The enlarged Ogg level algebra also fixes a canonical ordered primitive idempotent system. That order is transported across the residual-shell and affine four-sector algebras and induces a canonical ordered primitive basis on the rank-16 deck algebra. In this ordered form, the four-sector algebras become canonically identical as primitive-idempotent algebras. The same fixed line is then identified locally in the remaining realizations: as the charge-zero cubic sector, as the trivial real line in the real C₃ ⋊ C₂ form, as the h-axis in Koide coordinates, as the reversal-fixed opposite-pair class in the Apollonian quotient, and as 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. It builds on earlier published papers on full divisor locks, finite central-cover reductions, odd n-line linearization, support-completed diagonal-center coupling, Koide common reducts, Lorentzian fixed fibers, and the Star–Kneser defect theorem. This version makes the primitive basis structure fully explicit: the fifteen non-scalar primitive atoms are indexed by the Ogg primes, adjoining level 1 yields the sixteenth atom, and the enlarged algebra carries the canonical 1+1+7+7 decomposition together with the explicit Klein-four branch dictionary linking the invariant and exchanged sectors. Files included in this record: PDF preprint, LaTeX source, and source bundle.