bibkey: beluhov2026diamond authors: Nikolai Beluhov year: 2026 title: “Diamond Determinants and Somos Sequences” doi: null url: https://arxiv.org/abs/2602.24239v2 claim: “Conjecture 3: For every proper type 𝐧 of order n, it holds that dim Ω⊠ = ⌊n/2⌋.” strata_touched:

• D5/S0/Certificates/Combinatorics/GaleRobinsonKernelDimensionRefutation license: citation-only triage: anchor

Beluhov’s invariant-kernel dimension conjecture

Verified locator

URL: https://arxiv.org/abs/2602.24239v2

The quotations refer to the printed pages of arXiv:2602.24239v2. Conjecture 3, page 24, states:

For every proper type 𝐧 of order n, it holds that dim Ω⊠ = ⌊n/2⌋.

The surrounding paragraph on page 24 states:

For any proper type 𝐧, we can define the space ℰ as in Section 2; it is not too difficult to see that ℰ will still depend only on the parity of n.

Source definitions

Section 10, page 23:

Let 𝐧 = (n₁, n₂, n₃) with n₁, n₂, n₃ being positive integers such that n = n₁ + n₂ + n₃.

Section 10, page 24:

We call a type 𝐧 proper if it is primitive and n₁, n₂, n₃ are pairwise distinct.

Primitive means that the joint gcd of the entries is one. The quadratic recurrence form is

$$R_{\mathbf{n}}=a{x}_{n_1}{x}_{n_2+n_3}+b{x}_{n_2}{x}_{n_3+n_1}+c{x}_{n_3}{x}_{n_1+n_2}.$$

The coefficient field is the rational-function field in the three independent recurrence coefficients. The Lean representation is FractionRing (MvPolynomial (Fin 3) ℚ); rational-function fields formed from integer or rational polynomial coefficients are the same here.

Section 5, page 11:

This is equivalent to each exponent tuple (d₀, d₁, …, dₙ₋₁) which occurs in Φ satisfying d₀e₀ + d₁e₁ + ⋯ + dₙ₋₁eₙ₋₁ = e₀ + e₁ + ⋯ + eₙ₋₁ for all integer e ∈ ℰ.

The polynomials Φ which satisfy our additional constraint form a linear subspace Υ⊠ of Υ.

Let Ω⊠ be the kernel of φ over Υ⊠.

The source’s parity-only gauge bases on page 5 are 1,i for even order and i,i mod 2,(i+1) mod 2 for odd order. Thus the admissible degree-n monomials have exponent sum n and index-weight sum n(n−1)/2; odd order also requires even-index exponent sum (n+1)/2 and odd-index exponent sum n/2. These quotients are natural integer divisions.

The operator in Section 5, page 10 is

$${\phi }_t(\Phi )=x_0^{n-2}{R}_t\Phi -\Phi (x_0{x}_1,x_0{x}_2,\dots ,x_0{x}_{n-1},R_t).$$

Formal scope

The formal claim follows the source’s stated parity-only gauge reading. It quantifies over all proper types, using the K-dimension of the intersection of the admissible monomial span with the kernel of the displayed operator. The refutation uses type (1,3,6), of order 10, over the independent-parameter field K, and six independent kernel elements. It establishes a lower bound of six, rather than a matching upper bound.

The full gauge space determined by only the three nonzero terms of this type’s recurrence is a different interpretation. Its dimension and restricted kernel dimension are not proved by this module. Conjectures 1, 2 and 4–6 are outside the refutation’s formal conclusion. The source’s verified low-order results are not contradicted by this order-10 example.