Full Lean 4 verification, zero sorry stubs. Co-authored with Aristotle (Harmonic Math). E₈ paper in preparation.
Version 1.3: zenodo.org/records/1879...
— Chavez AI Labs | Applied Pathological Mathematics
"Better math, less suffering"
Screenshot of Lean 4 file header and Phase 4 completion summary generated by Aristotle (Harmonic Math). Shows Lean version 4.24.0 and Mathlib version f897ebcf. Phase 4 results: (1) Formally enumerated all 24 bilateral zero divisors as 24 unique index quadruplets supporting 48 signed pairs. (2) Verified family structure — proven Child_Q3Q2 is a bilateral zero divisor not in the Canonical Six; enumerated QxQ (3 types), PxP (4 types), PxQ (4 types) products. (3) Proven All_ZDs_Generated: every member of the 24-element family is generated by the Canonical Six. (4) Confirmed Canonical Six correspond to 5 of the 24 index quadruplets. Conclusion: the 24-element family is the closure of the Canonical Six under specified product generation rules.
Also proven: The Canonical Six are the minimal generating set for the complete 24-element bilateral zero divisor family. Every member is reachable from them via star/conjugation.
Screenshot of Lean 4 formal proof code. Theorem 1a (E₈ shell): proves all five P-vectors have squared norm 2 via native_decide +revert. Theorem 1b (Antipodal pair): proves v₂ + v₃ = 0 and simple reflection s_α₄ maps v₂ to v₃, via native_decide +revert. Theorem 1c (Single Weyl orbit): defines five explicit Weyl reduction sequences (seq1–seq5) proving all five vectors reduce to the dominant weight λ under the greedy Weyl reduction algorithm. Co-authored with Aristotle (Harmonic Math).
New in v1.3: P-vector images (5 distinct 8D coordinates) all satisfy ‖v‖² = 2 — E₈ first shell membership. All 5 form a single Weyl orbit with dominant weight ω₁. Antipodal pair v₂ + v₃ = 0, connected by simple reflection s_α₄.
Background: The Canonical Six work identically in both Cayley-Dickson AND Clifford algebras from 16D through 256D. They represent just 3.6% of all 168 sedenion zero divisors yet maintain exact structure through 256-fold complexity increase.
Schematic illustration of the five P-vector images of the Canonical Six bilateral zero divisor patterns projected onto the E₈ lattice first shell. Dark background with 240 white dots representing E₈ roots arranged on the first shell (‖v‖² = 2). Five labeled vectors — v₁ through v₅ — are plotted in three color groups: gold (v₁, v₄ — Color Group 1, Patterns 1 & 4), blue (v₂, v₅ — Color Group 2, Patterns 2 & 5), and red (v₃ — Color Group 3, Patterns 3 & 6). A double arrow connects v₂ and v₃, annotated "v₂ + v₃ = 0 (antipodal, s_α₄)". Note: 6 patterns produce 5 distinct P-vector images — Patterns 18 & 102 share the same geometric point. Lean 4 proven, zero sorry stubs. Chavez AI Labs. Schematic illustration credit: CAILculator MCP.
New result in higher-dimensional algebra:
The 6 framework-independent bilateral zero divisor patterns in 16D sedenions, aka The Canonical Six, have P-vector images that lie on the E₈ lattice first shell and form a single Weyl orbit. Lean 4 proven, zero sorry stubs.🧵
🎉Just hit 953 downloads!
Fun fact: 953 is a Sophie Germain prime (2×953 + 1 = 1,907 is prime too). Always inspired by her tenacity despite the barriers she faced.
v1.2 brings formal verification in Lean 4 for The Canonical Six
doi.org/10.5281/zenodo.17402495
#SophieGermain #Math #LeanProver #Algebra
877 downloads! Super prime as well as Pythagorean, Sexy, Cousin and Strong prime. Also, a rare Bell prime: 877 ways to partition a set of 7 elements.
Version 1.2 with Lean 4 formalization at 200 downloads!
Grateful to those who find sedenions interesting!
doi.org/10.5281/zenodo.18357723
#Math
For the #NumberTheory crowd - 797 is a beauty.
It’s a Super Prime: 139th prime (and 139 is prime),↔️ Two-Sided Truncatable (797, 97, 7 AND 79, 7 are all prime), Palindromic: 797,➕ Additive: 7+9+7 = 23 (Prime), Pythagorean: 11^2 + 26^2=797, and a Chen Prime.
#Math #Primes #OpenScience #Sedenions
797 downloads! Version 1.2 with Lean 4 formalization has 93% download-to-view ratio (136/146) just one week after update on Zenodo.
The Canonical Six zero divisor patterns compute from 16D sedenions to 256D in BOTH Clifford and Cayley-Dickson algebras.
doi.org/10.5281/zenodo.18357723
#Math #Lean4
709 downloads 🎯
V1.2 with Lean 4 formal verification surging: 63 downloads in 3 days, 96.9% conversion.
The Canonical Six gaining traction with rigorous proofs provided by Harmonic Math's Aristotle.
"Better math, less suffering."
doi.org/10.5281/zeno...
#Lean4 #FormalVerification #Mathematics
🎯 653 downloads - another Sophie Germain Prime!
Version 1.2: Lean 4 proofs formalize The Canonical Six across 16D-256D. Machine-verified certainty meets framework-independent elegance. ✅
Working independently, proving rigorously - l'esprit de Sophie Germain lives on.
doi.org/10.5281/zenodo.18357723
Fitting milestone for research on zero divisor patterns in sedenions that work identically in both Cayley-Dickson (non-associative) and Clifford (associative) algebras with dimensional persistence to 256D.
Formally verified via Lean 4 through Harmonic's Aristotle.
Rare numbers, rare structure. 🎯
563 downloads - third Wilson prime.
There are only THREE Wilson primes known in all of mathematics: 5, 13, and 563. These satisfy the condition (p-1)! ≡ -1 (mod p²), if a fourth exists, it's > 500,000,000. This will be my only Wilson prime download milestone.
doi.org/10.5281/zenodo.17574868
#Rare
500+ DOWNLOADS on Zenodo! 🎉📈
My paper on The Canonical Six crossed the 500-mark with a 71% conversion rate. Verified by #Lean4 with Harmonic Math's Aristotle. Moving high-dimensional algebra from "pathological" to predictable.
🔗 doi.org/10.5281/zeno...
#Math #ZeroDivisors #TwoAlgebras #Sedenions
431 downloads - Sophie Germain prime! (both p and 2p+1 are prime).
'Canonical Six' zero divisor patterns in sedenions that are framework-independent (both Cayley-Dickson and Clifford algebras) with dimensional persistence through 256D, formally verified via Lean 4.
doi.org/10.5281/zenodo.17574868
400 downloads of Zenodo paper on framework-independent zero divisor patterns in sedenion space with dimensional persistence to 256D.
Formally verified (Lean 4) by Harmonic Math's Aristotle, screenshot shows Aristotle proving in Cl(4,0) and Cl(5,0).
doi.org/10.5281/zenodo.17574868
#CliffordAlgebra
383 downloads. Landing on 383 is a mathematical "unicorn" moment as a rare Woodall Prime (6 * 2^6 - 1). Named for H.J. Woodall who studied them in 1917 with Allan J. C. Cunningham.
Also Palindromic Prime and Safe Prime.
doi.org/10.5281/zeno...
Formally verified by Harmonic Math's Aristotle (Lean 4).
349 downloads for another Prime milestone hit. 📈
The finding: 6/12 zero divisor patterns in sedenion space work identically in both non-associative Cayley-Dickson and associative Clifford algebras and scale to 256D. Lean 4 verified by Harmonic Math's Aristotle.
doi.org/10.5281/zenodo.17574868
Just reached 331 downloads of my math paper on "The Canonical Six".
Harmonic's Aristotle (Lean 4) has verified:
- All 6 framework-independent patterns in both CD4 and Cl(4,0)
- Dimensional persistence through 32D+
- CD-specific patterns provably fail in Clifford framework
doi.org/10.5281/zeno...
Just hit 300 downloads on Zenodo! 📈
Incredibly grateful to see this many people digging into framework-independent zero divisors in Sedenions and beyond.
📖 doi.org/10.5281/zeno...
Downloaders, let's talk! Would love to hear your thoughts and ideas about it.
#Math #ZeroDivisors #256D #Sedenions
271 downloads (passed 269 too fast to capture).
Twin primes 269 & 271 are partners, like zero divisor pairs in higher-dimensional algebras.
269 fits in 10 prime categories: Twin, Pythagorean, Strong, Eisenstein, Chen, Ramanujan, Good, Sexy, Pillai and Full Reptend.
doi.org/10.5281/zeno...
#Math
AI-generated purple and magenta fractal spiral, depicting the golden ratio/Fibonacci spiral pattern. Represents the mathematical beauty of the 233rd download milestone, which is both a Sophie Germain prime and Fibonacci prime.
233 downloads. Sophie Germain prime (2×233+1 = 467, prime) AND Fibonacci prime (13th in sequence).
Finding structure in dismissed spaces requires persistence, whether studying primes or exploring framework-independent patterns in higher-dimensional algebras.
doi.org/10.5281/zeno...
#NotNiceMath
Image shows poster of Apple TV show Pluribus on left with a screenshot of 223 downloads of Zenodo paper - a Carol Prime number. The post marks the 223rd download of the paper while offering background on Carol Primes and the TV show. Image courtesy of Apple TV+.
223 downloads. The Carol Prime: (2⁴-1)²-2 = 223. Only 7 known below a billion. Named by Cletus Emmanuel for his friend Carol Kirnon and their rarity reminds us of Carol Sturka in Pluribus taking on the hive mind of the whole planet. Some things remain distinct.
doi.org/10.5281/zeno...
#NotNiceMath
An image marking the 211 download milestone on Zenodo. A highlighted annotation explains the Chen Prime logic: 211 is prime, and 211 plus 2 equals 213, which is a semiprime (3 times 71). The image contrasts the 'nice' symmetry of 8D Octonions with the complex, pathological structure of 16D Sedenions, symbolizing a move toward mathematical truth over aesthetic elegance
211 downloads. A Chen Prime. p + 2 = 213, a semiprime (3 x 71). While the establishment polishes the "Nice Math" of the Octonions, we are exploring 16-dimensional sedenions and beyond. Shout-out to legendary Chinese mathematician Chen Jingrun. #NotNiceMath #Sedenions #Truth
doi.org/10.5281/zeno...
A Zenodo analytics screenshot showing 191 total downloads. The text identifies 191 as a Sophie Germain Prime and the post marks the project's expansion beyond octonions into 32-dimensional math space and beyond with Applied Pathological Mathematics.
191 downloads. A Sophie Germain Prime (2p + 1 = 383). We are inspired by the French mathematician and dare to venture with her perseverance beyond octonions with Applied Pathological Mathematics.
doi.org/10.5281/zeno...
#SophieGermain #Sedenions #FrontierMath #MathSky #SciSky #AIResearch #DeepTech