Infinite natural-basis quotient with finite full automorphisms

Kaygorodov, Martin Gonzalez and Viruel, Evolution algebras: a survey of results and a list of open problems, arXiv:2610.22248v1, section 2.2.2, Question 14 asks:

Describe the set of natural bases (up to permutation and scaling) of a non-degenerate evolution algebra. When is it finite? Can it be infinite for an algebra whose automorphism group is finite?

The endpoint is the entire final interrogative only. Its positive answer requires a permitted field and a finite-dimensional commutative bilinear, possibly nonassociative evolution algebra with zero annihilator, finite full linear multiplication-automorphism group, and infinitely many natural bases modulo all permutations and independent nonzero scalings. It does not ask for every field, a group-scheme classification, or algebra-automorphism orbits of bases.

A rank-one square can admit many orthogonal coordinate decompositions while its distinguished nonzero idempotent restricts multiplication automorphisms. These are different constraints on changes of natural basis.

The algebra $A_{8,1}$, arbitrary-natural-basis equations and finite full automorphism group are already in arXiv:1807.02362v1, Lemma 3.15 and section 5.1. The same full-group ingredient occurs in arXiv:2303.00862v1, Theorem 26 and Remark 29. The basis structure results of arXiv:2006.14460v1, Corollary 2.13 and Proposition 2.16, are also credited. The infinite quotient example in arXiv:2505.14584v1, Example 6.3, instead has nonzero annihilator and infinite full group. The application separates infinitely many classes in the exact quotient for the finite-group algebra.

Use $A=\mathbb C^2$ and $\mu((x,y),(z,w))=(xz+yw,0)$, a complex-bilinear product without conjugation. The standard basis is natural. Testing products against its two vectors shows $\operatorname{Ann}(A)=0$. The square is the first coordinate line, so this algebra is non-perfect as in the question's focus.

An arbitrary invertible complex-linear multiplication-preserving map fixes $(1,0)$, since its image is a nonzero idempotent. Its image of $(0,1)$ has first coordinate zero and second coordinate $d$ with $d^2=1$. Linearity therefore determines the whole map from $d\in{1,-1}$, giving an injection of the full group into Bool. No monomiality hypothesis is assumed.

For $t=n+1$, the vectors $(1,t),(-t,1)$ have zero mixed product and determinant $1+t^2\ne0$. An explicit inverse constructs their basis. If the first vectors for $t=n+1$ and $s=m+1$ are proportional without a swap, the scalar is one and $n=m$. With a swap, proportionality forces $ts=-1$, contradicting the positive natural integer $ts+1$ embedded in $\mathbb C$. This covers arbitrary complex nonzero scalars. The resulting map from the natural numbers into the exact quotient is injective.

Every basis in dimension two has two elements. Reindexing by Fin 2 uses Module.finrank_eq_card_basis and Module.Basis.reindex; changing that choice is exactly a permutation. The formal endpoint uses the canonical quotient of all natural Fin 2-indexed bases. No separate arbitrary-index equivalence is asserted to have been compiled.

A failure of bilinearity, finite dimension, zero annihilator, actual basis independence and spanning, full-group finiteness, or separation under any allowed permutation and independent nonzero scaling invalidates the result. Infinitely many ordered representatives or a finite monomial subgroup alone would not answer the question. Quotienting further by algebra automorphisms would change the target.

The source endpoint is D5/S3/QuadraticForms/EvolutionAlgebras/FiniteAutomorphismsInfiniteNaturalBases.result. Its exact type existentially quantifies a bilinear product on $\mathbb C^2$ and conjoins dimension two, commutativity, zero annihilator, an actual natural basis, finite full automorphisms and an infinite exact quotient.

What the result shows

• The complete final interrogative has an affirmative answer over $\mathbb C$.
• The mechanism is the distinction between orthogonal natural-basis changes and multiplication automorphisms fixing the distinguished idempotent.
• No finite enumeration or numerical experiment supports the conclusion; its infinite family and all automorphisms are handled symbolically.
• The first two sentences' general description and finiteness classification remain outside this result.

The bounded searches found no prior published whole answer. This is not a worldwide absence or priority claim. Authenticated code search was unavailable, and no search-complete status is claimed. The old algebra, classification and basis ingredients retain their source credit.