Framework-Independent Zero Divisor Patterns in Higher-Dimensional Cayley-Dickson Algebras: Discovery and Verification of The Canonical Six

We report the discovery of 12 zero divisor patterns in 16-dimensional sedenion space exhibiting dimensional persistence and framework-dependent behavior, with six patterns demonstrating framework independence across both Cayley-Dickson and Clifford algebraic constructions. Computational verification across five Cayley-Dickson dimensions (16D through 256D) and two Clifford algebra dimensions (16D and 32D) demonstrates exact preservation of zero divisor properties through successive dimensional doublings, achieving machine precision (≈ 10−15) in all successful tests. The patterns divide equally: six universal structures (The Canonical Six) maintain zero divisor properties in both associative (Clifford) and non-associative (Cayley-Dickson) frameworks, while six construction-dependent patterns succeed only in Cayley-Dickson algebras, revealing a fundamental 50/50 split. Representing only 3.6% of all 168 sedenion zero divisors, these framework-independent patterns exhibit superlinear dimensional stability—maintaining exact structure through 256-fold complexity increase—suggesting they occupy mathematically distinguished positions indicative of deeper organizational principles in higher-dimensional algebras and challenging the characterization of these algebras as pathological. Full formal verification in Lean 4 confirms all core structural claims with zero sorry stubs. New results in v1.3 establish that the P-vector images of the Canonical Six lie on the E8 lattice first shell and form a single Weyl orbit, and prove that the Canonical Six constitute the minimal generating set for the complete 24-element bilateral zero divisor family.