The Quaternionic Kernel Theorem: Third-Generation Decoupling from H ⊂ O

We establish, in a computer-assisted manner for a fixed octonion multiplication convention, that the vacuum kernel mechanism for third-generation decoupling in the octonion flavor framework is controlled by a unique quaternionic subalgebra H ⊂ O, selected by the Cayley–Dickson doubling implicit in the multiplication table. Although every Fano line generates a copy of H, exhaustive computation across all seven lines and all seven vacuum choices shows that only the line L = {2,4,6} satisfies the kernel theorem: for every vacuum P_k with k in the complementary quadrangle Q = {1,3,5,7}, the kernel-active indices are exactly L. This identifies H_L = span{1,e_2,e_4,e_6} as the canonical Cayley–Dickson subalgebra, and explains the algebraic origin of third-generation decoupling in the octonion flavor framework. Files included.- quaternionic_kernel_theorem.pdf: article- flavor_from_octonions.py: unified companion script for the octonionic flavor program- run_qkt.sh: one-command reproducer- requirements.txt: dependencies- README.md: usage notes Quick reproduction.pip install -r requirements.txtbash run_qkt.sh Equivalent command:python3 flavor_from_octonions.py --section 6 --verbose