slug: kmv27-evobalance
bibkey: kaygorodov2026evolution
doi: 10.48550/arXiv.2610.22248
url: https://arxiv.org/abs/2610.22248v1
triage: theorem
motivation_gids:
  - D5/S3/QuadraticForms/EvolutionAlgebras/DegenerateGraphBalanceRefutation

Natural-basis dependence of balance in degenerate evolution algebras

Problem

Kaygorodov, Martín González and Viruel, Evolution algebras: a survey of results and a list of open problems, arXiv:2610.22248v1, Question 27, page 24, ask:

Is the balance independent of the natural basis for arbitrary (possibly degenerate) evolution algebras?

The source’s section 1.1 states: “Unless stated otherwise, algebras are finite-dimensional.” The assertion concerns every field, bilinear evolution multiplication and pair of natural bases. It adds no nondegeneracy, perfectness, characteristic or connectedness assumption. The other two questions about a complete system of graph invariants are outside this settlement. Preregistration: #21625.

Statement echo

The graph definitions also accept infinite vertex types and the height lemma arbitrary moduli; these stronger helper domains do not restrict the finite-basis assertion. Closing universes at Type supplies a closed proposition and includes the rational counterexample. The zero gcd is zero.

Motivation

An annihilator direction permits a shear of a natural basis without changing any mixed product. Square-support coordinates can nevertheless change. Balance therefore detects information that this freedom need not preserve.

Gap

Alshatnawi, Costoya and Viruel, arXiv:2607.15154v1, Definitions 2.1, 2.2 and 2.4, supply the directed-path and all-cycle balance definitions. Their Theorem 6.1 assumes a nonzero nondegenerate evolution algebra. The survey’s perfect-algebra rigidity theorem has a separate stronger hypothesis. Neither assertion covers the degenerate example. The literature screen in #21625 located no exact settlement within its inspected scope; completeness and priority are ASSUMED-UNVERIFIED. The two source DOIs have distinct canonical notes: Library/QuadraticForms/kaygorodov2026evolution.md and Library/QuadraticForms/alshatnawicostoyaviruel2026grothendieck.md.

Route

Over any field take a basis $(a,b,u,z)$ with $a^2=b$, $b^2=a+u$, $u^2=z$, $z^2=0$, and zero mixed products. The new basis $(a+z,b,u,z)$ is obtained by a transvection. Its only changed square row is $b^2=(a+z)+u-z$. The coefficient $-1$ stays nonzero in characteristic two.

The old graph has arrows $0\to 1$, $1\to 0$, $1\to 2$, $2\to 3$. Heights $(0,1,0,1)$ in $\mathbb{Z}/2\mathbb{Z}$ increase along every arrow. On any closed signed path the height increments telescope, so every balance is even. The directed two-cycle has balance two, proving that the all-cycle gcd is two. The new graph adds $1\to 3$; the path $1\to 2\to 3\leftarrow 1$ has balance one, so the new gcd is one. The private arbitrary-field helpers old_natural, new_natural, evolution, old_graph and new_graph verify the algebra and coordinate transport. old_balance and new_balance supply the two gcd values. The public result specializes to $\mathbb{Q}$ and contradicts the universal equality.

Falsifier

The assertion would not be refuted if either displayed family failed to be a basis, had a nonzero mixed product, used column rather than row support, or replaced the all-cycle gcd by a bounded-length calculation. The kernel proof checks basis transport, naturality, exact support and the all-length balance argument. The loop ambiguity in the source’s informal wording has no effect on these loop-free graphs.

Evidence

The settling declaration is D5/S3/QuadraticForms/EvolutionAlgebras/DegenerateGraphBalanceRefutation.result, with literal type Not claim. Its module contains no other authored public theorem. The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9). The module’s admission basis is open-problem-resolution (#21625; Refuted); all theorem proof shapes are conservatively bind-only, with every private helper consumed. No escape-witness admission is claimed.

Exact finite computation

Experiment directory at commit 229beb623231963298f417d3dec1bbb44b3eeab0. The pinned check.py was extracted from that commit and executed with python3 docs/reports/kmv-q27-balance/check.py, exit 0; SHA-256 a8d0058590501bd93df3cc66c236eb1f3f603cc87d56fa66c2507ad8fccb6663. The final result is ALL_CHECKS_OK. Exact rational and modular arithmetic checks 95 cases: $m=2,\dots ,20$ over $\mathbb{Q},{\mathbb{F}}_2,{\mathbb{F}}_3,{\mathbb{F}}_5,{\mathbb{F}}_7$. It checks all basis products, determinant one, annihilator dimension one, exact support, connectedness, absence of loops, height increments and balances $m,1$. A spanning-forest potential calculation gives the all-cycle gcd for each tested graph. A separate enumeration cross-checks lengths at most seven for $m=2,\dots ,6$; that bounded enumeration does not establish an unbounded statement. These are computed readings, separate from the kernel proof.

Triage

Coverage

What the settlement shows

Proved in prose, uniform family. For every field and integer $m\ge 2$, use a basis $(c_0,\dots ,c_{m-1},u,z)$ with $c_i^2=c_{i+1}$ for $i<m-1$, $c_{m-1}^2=c_0+u$, $u^2=z$, $z^2=0$, and zero mixed products. If a vector annihilates this algebra, multiplying it by each $c_i$ forces every $c_i$ coefficient to vanish, since each $c_i^2$ is nonzero; multiplying it by $u$ forces the $u$ coefficient to vanish. Conversely $z$ annihilates every basis vector. Thus the annihilator is exactly the line spanned by $z$.

Replacing $c_0$ by $c_0+z$ has inverse obtained by subtraction of $z$, and preserves naturality. In the changed coordinates $c_{m-1}^2=(c_0+z)+u-z$, so exactly the arrow $c_{m-1}\to z$ is added. For the old graph assign $h(c_i)=i$, $h(u)=0$, $h(z)=1$ in $\mathbb{Z}/m\mathbb{Z}$. Every arrow raises height by one. The sum of signed height differences on a closed path is zero, so $m$ divides every cycle balance. The core directed cycle has balance $m$, hence the gcd is $m$. In the new graph $c_{m-1}\to u\to z\leftarrow c_{m-1}$ has balance one, hence the gcd is one. This proves the uniform mathematical family in prose, without claiming a uniform Lean formalization.

Mechanism and boundary. The annihilator shear preserves all mixed-product relations but changes square support. The same algebra can therefore have arbitrarily large balance in one natural basis and balance one in another. Nondegenerate balance invariance and perfect-algebra basis rigidity keep their original hypotheses. No consequence requiring the degenerate extension can use that extension as a universal invariant. The other questions in Question 27, including a complete invariant system for nondegenerate algebras, remain open here.

ASSUMED-UNVERIFIED

The literature screen’s completeness and absence of another published settlement are ASSUMED-UNVERIFIED. The screen reported in #21625 inspected both v1 sources, related papers and author searches; citation enumeration was limited by OpenAlex and Semantic Scholar HTTP 429 responses, and broad issue searches reached their caps. No exhaustive priority claim is made. The entire $m\ge 2$ family has a prose proof but no Lean proof in this module.