Flavor from Octonion Geometry: From Cayley–Dickson Structure to CKM and PMNS Mixing

We show that a large part of quark–lepton flavor structure — one-generation decoupling, mixing textures, and the CKM/PMNS dichotomy — can be traced to rigid algebraic features of the complexified octonions C⊗O, organized by the Cayley–Dickson bipartition O = H ⊕ H·e and a primitive idempotent vacuum. **Central result. ** The Quaternionic Kernel Theorem (QKT): given a fixed multiplication-table convention, the vacuum eigenspace V⁺ = X: X·s = X uniquely selects a quaternionic subalgebra HL ≃ H (equivalently, a unique Fano line L = 2, 4, 6) characterized by eigenspace-preservation under left multiplication. This selection is vacuum-induced: different vacua are G₂-conjugate, while the structural content — existence and uniqueness of the selected subalgebra — is invariant. The proof combines a structural argument (Cayley–Dickson parity + an orthogonal triple criterion in Im (O) ) with exhaustive verification in exact Q (i) arithmetic. **Downstream consequences (no fitted parameters): **- One generation index (k=6) belongs to the selected subalgebra and decouples at leading order (rank Y⁽⁷⁾ = 2), while the other two (k=3, 5) remain active. - A three-stage vacuum kernel mechanism (bottleneck → dagger projection → annihilation) explains the row vanishing structurally. - The CKM/PMNS dichotomy reduces to a discrete leakage pattern among a finite, fully classifiable set of block pairs. - Geometric invariants λ = 1/ (2√5) ≈ 0. 224 (PDG: 0. 225) and A = √ (2/3) ≈ 0. 817 (PDG: 0. 826) emerge from the projection chain. - The division-algebra tower terminates at O by a rigidity argument (Hurwitz obstruction + sedenion collapse diagnostic). **Epistemic convention. ** The article distinguishes three levels: (1) Theorem — purely algebraic proof; (2) Theorem (computer-assisted) — structural proof + exhaustive exact verification; (3) Observation — empirical pattern. All claims are explicit about their status. **This deposit contains: **- The article (7 pages, RevTeX two-column format) - LaTeX source- A companion Python script (~1100 lines) that reproduces every computer-assisted result in exact Q (i) arithmetic (Sections 1–7) and numerical phenomenology (Sections 8–10) - README with usage instructions and section–theorem correspondence table This is a unified presentation consolidating and extending results from a series of companion notes (CKM, PMNS, triality, vacuum kernel, quaternionic kernel, Fano selection, lifting, projection principle, tower termination) previously deposited on Zenodo.