No octonion push-forward for the exceptional Vidinli algebra

Abstract

The exceptional seven-dimensional Vidinli product is not a linear push-forward of octonion multiplication.

Definition 1.1 (The exceptional Vidinli product).

$$\forall a:\mathrm{Fin}\left(7\right)\to \mathbb{R},\forall b:\mathrm{Fin}\left(7\right)\to \mathbb{R},\mathrm{vidinliMul}\left(a,b\right)=(\lambda i:\mathrm{Fin}\left(7\right)\mapsto \mathrm{ite}\left(i=0,a\left(0\right)\cdot b\left(0\right)-\sum _{j\in Finset.Ioi\left((0:\mathrm{Fin}\left(7\right))\right)}(a\left(j\right)\cdot b\left(j\right))+(a\left(1\right)\cdot b\left(2\right)-a\left(2\right)\cdot b\left(1\right))+(a\left(3\right)\cdot b\left(4\right)-a\left(4\right)\cdot b\left(3\right))+(a\left(5\right)\cdot b\left(6\right)-a\left(6\right)\cdot b\left(5\right)),a\left(0\right)\cdot b\left(i\right)+b\left(0\right)\cdot a\left(i\right)\right))$$

Formalization. D5/S3/HomologicalAlgebra/CoskunEdenVidinliOctonion.vidinliMul (✓ std3).

Citation. O. Coşkun and A. Eden (2025). Structure of Vidinli Algebras: Simple Nonassociative Jordan–Lie Algebras with Azumaya Properties. DOI: 10.48550/arXiv.2511.09395. URL: https://arxiv.org/abs/2511.09395v1.

Commentary.

Definition 5.1 and its coordinate expression, pp. 19–20: “a ⊻₇ b = c₁e₁ + ∑{i=2}^{7} cᵢeᵢ, where c₁ = a₁b₁ − ∑{i=2}^{7} aᵢbᵢ + (a₂b₃ − a₃b₂) + (a₄b₅ − a₅b₄) + (a₆b₇ − a₇b₆), cᵢ = a₁bᵢ + b₁aᵢ for i = 2, …, 7.” The coordinate i : Fin 7 represents the paper’s coordinate i + 1. Finset.Ioi (0 : Fin 7) consists of indices 1 through 6.

Definition 1.2 (The octonion clause of Remark 5.9).

$$(\mathrm{claim}:\mathrm{Prop})=(\exists \nu :(\mathrm{Fin}\left(7\right)\to \mathbb{R}){\to }_l[\mathbb{R}](D5.S3.VertexAlgebra.MonsterTwistedOctonionNorm.Octonion),\exists \pi :(D5.S3.VertexAlgebra.MonsterTwistedOctonionNorm.Octonion){\to }_l[\mathbb{R}](\mathrm{Fin}\left(7\right)\to \mathbb{R}),\forall a:\mathrm{Fin}\left(7\right)\to \mathbb{R},\forall b:\mathrm{Fin}\left(7\right)\to \mathbb{R},\mathrm{vidinliMul}\left(a,b\right)=\pi \left(D5.S3.VertexAlgebra.MonsterTwistedOctonionNorm.mul\left(\nu \left(a\right),\nu \left(b\right)\right)\right))$$

Formalization. D5/S3/HomologicalAlgebra/CoskunEdenVidinliOctonion.claim (✓ std3).

Citation. O. Coşkun and A. Eden (2025). Structure of Vidinli Algebras: Simple Nonassociative Jordan–Lie Algebras with Azumaya Properties. DOI: 10.48550/arXiv.2511.09395. URL: https://arxiv.org/abs/2511.09395v1.

Commentary.

Remark 5.9, p. 23: “By [5, Proposition 9], see also Section 3.1, the Vidinli multiplication in dimension three arises as a push-forward of quaternion multiplication. Whether a similar push-forward description exists for the exceptional Vidinli algebra V₇ in terms of the octonions, or more generally for V_{2^k−1} in relation to higher Cayley-Dickson algebras, remains an open question.” The proposition expresses the affirmative answer to the octonion clause: there are real-linear maps ν and π satisfying the product identity for every pair a, b. The octonions and their multiplication are the eight-coordinate signed Cayley–Dickson model MonsterTwistedOctonionNorm.Octonion and MonsterTwistedOctonionNorm.mul. Neither map is assumed to preserve products.

Theorem 1.3 (The octonion clause has a negative answer).

$$\neg (\exists \nu :(\mathrm{Fin}\left(7\right)\to \mathbb{R}){\to }_l[\mathbb{R}](D5.S3.VertexAlgebra.MonsterTwistedOctonionNorm.Octonion),\exists \pi :(D5.S3.VertexAlgebra.MonsterTwistedOctonionNorm.Octonion){\to }_l[\mathbb{R}](\mathrm{Fin}\left(7\right)\to \mathbb{R}),\forall a:\mathrm{Fin}\left(7\right)\to \mathbb{R},\forall b:\mathrm{Fin}\left(7\right)\to \mathbb{R},\mathrm{vidinliMul}\left(a,b\right)=\pi \left(D5.S3.VertexAlgebra.MonsterTwistedOctonionNorm.mul\left(\nu \left(a\right),\nu \left(b\right)\right)\right))$$

Proof. Machine-checked in Lean as D5/S3/HomologicalAlgebra/CoskunEdenVidinliOctonion.result (✓ std3). ∎

Resolves. Problems/coskun-eden-2025-vidinli-octonion-pushforward (refuted) by D5/S3/HomologicalAlgebra/CoskunEdenVidinliOctonion.result.

Source. Repository-derived.

Acknowledgement. O. Coşkun and A. Eden (2025). Structure of Vidinli Algebras: Simple Nonassociative Jordan–Lie Algebras with Azumaya Properties. DOI: 10.48550/arXiv.2511.09395. URL: https://arxiv.org/abs/2511.09395v1.

Commentary.

The right unit forces ν to be injective and π to be surjective. Rank-nullity gives a one-dimensional kernel for π. Left multiplication by the nonzero octonion ν(e₂) is injective by the multiplicative positive-definite square norm. The vectors e₂ + e₃, e₄, e₅, e₆, e₇ are independent and are killed by left multiplication by e₂ in the Vidinli algebra. Their images would therefore embed a five-dimensional real vector space into ker π, contradicting rank-nullity. Here e₂ in the paper is Pi.single 1 1 in the zero-based coordinates. The statement concerns arbitrary real-linear maps. It does not address the higher Cayley–Dickson clause, where zero divisors prevent this injectivity argument.

References

• Truth anchor: D5/S3/HomologicalAlgebra/CoskunEdenVidinliOctonion.claim
• Truth anchor: D5/S3/HomologicalAlgebra/CoskunEdenVidinliOctonion.result
• Truth anchor: D5/S3/HomologicalAlgebra/CoskunEdenVidinliOctonion.vidinliMul
• Dependency: D5/S3/VertexAlgebra/MonsterTwistedOctonionNorm