A codimension-one minimal-perfect criterion
Abstract
An exact all-parameter nilpotent and cyclic criterion for minimal perfect algebras with a square-zero hyperplane.
Definition 1.1 (The square-zero hyperplane product).
Formalization. D5/S3/QuadraticForms/EvolutionAlgebras/HuEvolutionCodimensionOneCriterion.codimOneProduct (✓ std3).
Source. Repository-derived.
Commentary.
Coordinate zero is the quotient vector u; coordinate i.succ is the i-th coordinate of W. Here tail(x)(j)=x(j.succ). The law is u squared = au+c, uw=wu=Tw and W squared =0. Matrices act on column vectors. There is no assumption of associativity, a unit, a natural basis or a Jordan identity.
Theorem 1.2 (The complete nilpotent and cyclic equivalence).
Proof. Machine-checked in Lean as D5/S3/QuadraticForms/EvolutionAlgebras/HuEvolutionCodimensionOneCriterion.codimension_one_criterion (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. Xing-Yu Hu; Ran Wen (2026). Idempotent-free non-solvable evolution algebras over C. DOI: 10.48550/arXiv.2609.25023. URL: https://arxiv.org/abs/2609.25023v1.
Commentary.
The criterion and its source proof are due to Wenlin Zhang and Haobo Ma, Structure tests for stable derived terms, section 2, criterion (4) and identity (5). The authored source has SHA256 1051e84fd643f3c3c528c8eb000fcecd2796054aadf6d2aa855815ad91192222. No public-priority claim is made. Hu and Wen, arXiv:2609.25023v1, Theorem 2.1, Lemma 2.3 and Proposition 2.4 include the boundary example m=1, a=2, T=1, c=−1 at their parameter 2.
García-Martínez and Pérez-Rodríguez, DOI 10.1007/s00013-026-02251-0, Theorems 2.1 and 3.5, and Ladra and Pérez-Rodríguez, DOI 10.5802/crmath.804, Theorem 2 and Corollary 4, treat regular evolution algebras with natural bases. Kaygorodov and Volkov, arXiv:1710.08943v2, section 4, Lemma 4.1, impose generation type one. Those additional hypotheses are absent here; none of these results is used as an arbitrary-subspace minimality criterion.
In the displayed formula, 1 is the m by m identity matrix. The matrix krylov(N,c,m) has entry (i,j) equal to ((N raised to j.val).mulVec c)(i). Its columns are c,Nc,…,N raised to m−1 times c, in that order. The exponent is exactly m. Arbitrary c, including zero, and every complex subspace are included. The square of a subspace means the span of all pair products. Minimal perfect means V squared = V and every nonzero complex subspace U satisfying U squared = U equals V.
A nonzero perfect subspace contains a normalized vector u+z. Its intersection S with W is invariant under T and N, and contains c+2Nz. Cyclicity puts W inside S+NW. Repeated lifting gives W=S+N raised to m times W=S, so the subspace is the whole algebra.
Conversely, an invariant subspace S containing c+2Nz gives the subspace generated by u+z and S. It is closed under multiplication. Left multiplication by u+z is injective because a and T are invertible; its restriction is therefore surjective, proving equality with its full product span. The Fitting generalized-zero component supplies a proper such subspace when N is not nilpotent. When N is nilpotent but c is not cyclic, its finite orbit span supplies one instead.
The arbitrary-operator-to-Jordan product-preserving isomorphism, the idempotent obstruction, square-zero extensions and Hu–Wen Question 2.11 are separate assertions and do not follow as completion claims here.