Power sums of the zeros of the BLZ polynomials: a proof of the second-power-sum conjecture of Masoero and Ruzza, a closed formula for the third power sum, and the membership half of their Question 3.7
Abstract
For a partition λ and a parameter β, Masoero and Ruzza (arXiv:2605.24563v2) attach to the Laguerre Wronskian Φ^((β))_(λ,λ') a monic polynomial P^((β))_λ of degree |λ| — the BLZ polynomial whose zeros zᵢ^((β))(λ) are the poles of the monster potentials of Bazhanov–Lukyanov–Zamolodchikov at the free-fermion point — and prove that the sum of the zeros is the shifted symmetric function 2p4-2p2²+3βp3+β²p2+1/2p2 of λ (their Proposition 3.6). For the sum of the squares they conjecture a degree-six shifted symmetric expression (their Eq. (3.8)), "verified for all λ ∈ ℙ with |λ| ≤ 15", and they ask (Question 3.7) whether the k-th power sum lies in Λ^(*)[β]_(≤ 2k+2) for every k. We prove the conjecture for every partition and every β. In the reversed Wronskian matrix the coefficient polynomial Gₚ(X)=(-1)ᵖ ff((X/2), p) ff((X/2-β), p)/p! attached to a column is literally the same for the two kinds of Laguerre column, so the determinant is alternating in the nodes, the Vandermonde divides it, and the k-th subleading coefficient is a symmetric polynomial in the nodes of total degree at most 2k. This places the difference of the two sides of Eq. (3.8) in Λ^(*)[β]_(≤ 16), a space pinned down by the partitions of size at most 16: a rank-231 certificate and the exact verification of Eq. (3.8), as an identity of polynomials in β, for every partition of size at most 16 finish the proof. The proof itself is on paper; what is machine-checked is exactly its two finite ingredients — the rank certificate and Eq. (3.8) for |λ| ≤ 16 — together with Eq. (3.8) for |λ| ≤ 17 and, at five values of β that decide it, for |λ| ≤ 21. The same lemma settles, again on paper, the membership half of Question 3.7, Σᵢ (zᵢ^((β)))ᵏ ∈ Λ^(*)[β]_(≤ 8k) for every k; only the sharp bound 2k+2 remains open. We also give a closed formula for the third power sum, an element of Λ^(*)[β]_(≤ 8) that the source does not have, found by exact fitting on |λ| ≤ 10, verified on all 369 partitions of size 11 to 14 before being written down, and machine-checked for |λ| ≤ 14; and we record that the top-degree layer of the k-th power sum is row 2k of OEIS A088617 reversed, with the Catalan number C₂ₖ in front of p(2k+2), for k=0,1,2,3.
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Source snapshot 2026-09-07 03:53 UTC
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1601127c124ee515b465559dcb63e1bcb6c5bbac21b5199235f3e04548f64542
Claim ledger
Stated results
BZ1known2026-09-07
The source's PROVED Proposition 3.6 (Eq. (3.5)), sumᵢ zᵢ^((beta))(lambda) = 2p₄ - 2p₂² + 3 beta p₃ + beta² p₂ + p₂/2, re-derived from the Laguerre Wronskian itself as an identity of polynomials in beta, for every partition with |lambda| <= 16; together with the Vandermonde control phi₀ = 6144^N prod_(i<j)(Xⱼ - Xᵢ), i.e. the source's kappa of Eq. (2.8), asserted at every one of the 54,684 evaluations.
BZ2known data2026-09-07
The source's conjectural Eq. (3.8) for sumᵢ (zᵢ^((beta)))², verified as an identity of POLYNOMIALS IN beta (by evaluation at 2rs+5 half-integer points, with the degree bound deg_beta phiₖ <= rs+k proved in the record) for every partition with |lambda| <= 15 – the range the source itself reports.
BZ3candidate2026-09-07
Eq. (3.8) as an identity of polynomials in beta for every partition of 16 and of 17 – past the range the source reports (|lambda| <= 15). 297 + 231 partitions, 41,632 determinant evaluations, kernel-bound.
BZ4candidate2026-09-07
Eq. (3.8) and Proposition 3.6 at beta in 1/2, 3/2, 5/2, 7/2, 9/2 for every partition of n, 18 <= n <= 21 (2294 partitions); by the source's Theorem 2.2 (deg_beta of both sides at most 4) five points decide the identity, so this is the full conjecture at those partitions. Exact Python extends the same check to |lambda| <= 25 (9,295 partitions).
BZ5routine2026-09-07
The finite-determination certificate: the 231 monomials pₘu (mu with all parts >= 2, |mu| <= 16) spanning Lambda*_(<=16) have rank 231 over F_(2³1-1) against the first 509 partitions of size at most 16, so a shifted symmetric function of degree at most 16 vanishing on every partition of size at most 16 is zero. 509 is minimal (rank 230 on 508 columns, and 508 = sum_(n<=14) p(n) exactly, so the 509th column is the first partition of 15).
BZ6prose2026-09-07
THEOREM (proved in the record, not formalized): the source's conjecture Eq. (3.8) is TRUE – for every partition lambda and every beta, sumᵢ (zᵢ^((beta))(lambda))² equals the stated degree-6 shifted symmetric expression.
This ledger entry is reported in prose and is not bound to a Lean theorem.BZ7candidate2026-09-07
A NEW closed formula for the THIRD power sum sumᵢ (zᵢ^((beta))(lambda))³, an element of Lambda*[beta]_(<=8), kernel-checked for every partition with |lambda| <= 14 at the seven points beta = 1/2, 3/2,..., 13/2 (which by the source's Theorem 2.2 decide the identity, both sides having beta-degree at most 6). It answers the k=3 case of the source's Question 3.7 affirmatively, at the predicted weight 2k+2 = 8.
BZ8routine2026-09-07
Kernel-clean (by ring over Q, no native_decide): the source states its conjecture twice, and the two statements are the same expression. Eq. (3.5) equals its Corollary 7.7 form 2 I₃ - 2 I₁², and Eq. (3.8) equals its Remark 7.8 form (56/3) I₅ - 40 I₃ I₁ + (64/3) I₁³ + 12 I₃ - 12 I₁², in the quantum-KdV eigenvalues I₁, I₃, I₅ of Theorem 7.6.
BZ9routine2026-09-07
Negative controls and non-vacuity: the partition generator reproduces p(n) for n <= 12; every partition of 6 is tested at at least 7 values of beta; the shifted symmetric data is nonzero; deleting the beta⁴ p₂ term or the (27/8) p₂ term from Eq. (3.8), or perturbing 14 p₆ to 15 p₆, each breaks the identity already at |lambda| <= 4 (too-small refutation); the same expression built from Phi_(lambda,mu) holds for mu = lambda' and FAILS for each of the other four partitions of 4 at lambda = (3,1) (too-large refutation); and the rank certificate of BZ5 is sharp (rank 230 on 508 columns).
BZ10prose2026-09-07
THEOREM (proved in the record, not formalized): the MEMBERSHIP half of the source's Question 3.7. For every k, sumᵢ (zᵢ^((beta))(lambda))ᵏ lies in Lambda*[beta]_(<=8k); only the sharp bound 2k+2 the source asks for remains open.
This ledger entry is reported in prose and is not bound to a Lean theorem.BZ11candidate2026-09-07
OBSERVATION: the top-degree layer of sumᵢ (zᵢ^((beta)))ᵏ, i.e. the coefficients of (p₂ₖ₊₂, beta p₂ₖ₊₁,..., beta²ᵏ p₂), is row 2k of OEIS A088617 reversed, T(n,j) = C(n+j,n) C(n,j)/(j+1) (Schroeder paths; row sums the large Schroeder numbers A006318). Confirmed at k = 0 (1), k = 1 (2,3,1 = row 2 reversed), k = 2 (14,35,30,10,1 = row 4 reversed) and k = 3 (132,462,630,420,140,21,1 = row 6 reversed, a seven-coefficient prediction made before the fit of BZ7 was run, all seven correct). In particular the leading coefficient is the Catalan number C₂ₖ.
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source. D. Masoero, G. Ruzza, *ODE/IM Correspondence at the Free-Fermion Point. Laguerre Wronskians, Shifted Symmetric Functions, and Quantum KdV*, arXiv:2605.24563 v2 (submitted 23 May 2026, v2 of 2 Jun 2026, comment line "V1: 45 pages; V2: 45 pages, minor corrections"; primary math-ph, cross-listed hep-th and math.CA). The local corpus holds v1; everything in this family was read from the compiled v2 e-print (tectonic on the source's own Laguerre_V2.tex; theorem, equation and page numbers taken from the resulting.aux, never counted f
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- 2026-09-07 03:53 UTC
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