Formalizing Zero Divisor Patterns in 16D to 256D

Stream: new members

[Paul Chavez]

Hi everyone. I am Paul Chavez and I'm working on formalizing framework-independent zero divisor patterns that exhibit dimensional persistence from 16D sedenions to 256D.

I have identified six patterns that can be calculated in both Cayley-Dickson and Clifford algebras in 16D that I am calling the 'Canonical Six' because the same patterns exist in 32D, 64D, 128D and 256D. I have a preliminary Lean 4 formalization via Harmonic Math's Aristotle and am looking to optimize it for potential mathlib compatibility.

Zenodo record for reference: https://doi.org/10.5281/zenodo.17574868

Thank you.

[Violeta Hernández]

I'm a bit confused on what you did, so please correct if I'm wrong: you searched for specific indices in Cayley-Dickson algebras such that (e_i ± e_j) * (e_k ± e_l) = 0?

[Violeta Hernández]

Assuming this is what you did, I really don't think it's notable, or Mathlib worthy. First of all, the specific bases that you get out of applying the Cayley-Dickson construction are far from unique, and these structures have many automorphisms, meaning that the exact indices bear little significance. But also, the nature of the Cayley-Dickson construction means that each algebra embeds cleanly into the next: e_1 through e_16 in the 16-nions are the same as e_1 through e_16 in the 32-ions, etc. So it's not surprising that these equations hold across all of these.

[Paul Chavez]

Hi Violeta,

Thanks for engaging! I think my intro wasn't clear and there's been a misunderstanding about the core claim. The core claim isn't about patterns within Cayley-Dickson algebras alone.

You're absolutely right that finding zero divisor patterns purely in CD wouldn't be notable - automorphisms and embedding structure would explain dimensional persistence there.

The key finding is framework independence**:

I identified 12 zero divisor patterns in Cayley-Dickson sedenions (CD4). When I tested these same index patterns in Clifford algebras:

• 6 patterns produce zero divisors in BOTH frameworks
• 6 patterns work in Cayley-Dickson but provably FAIL in Clifford

These are fundamentally different algebraic structures:

• Cayley-Dickson: e_i² = -1, non-associative
• Clifford: e_i² = +1, associative

Yet 6 patterns transcend this - same indices, same zero products, different multiplication rules.

The dimensional persistence within CD (which you correctly note isn't surprising) extends to Clifford: the same 6 patterns also work in Cl(4,0) and Cl(5,0).

Aristotle formalized this in Lean 4 (822 lines) - proving the patterns work in both CD4 and Cl(4,0), and that the CD-specific patterns fail in Clifford.

This cross-framework structure is what I'm calling "framework independence."

Does this clarify? I'm curious if you see mathematical significance in patterns that transcend construction methods this way, or if there's something I'm missing about why this would be expected.

Best,

Paul