Explicit kernel elements for multiplication by a general linear form on Bbbk[x₁,…,xₘ]/(x₁²,…,xₘ²,e²), and spanning sets for the kernel on the lines m=3d-3-k
Abstract
Let Bbbk have characteristic zero, let E=Bbbk[x₁,…,xₘ]/(x₁²,…,xₘ²), let e=x₁+…+xₘ, and let A=E/(e²), so that Aⱼ=Eⱼ/e²Eⱼ₋₂. Crispin Quiñonez, Lundqvist and Nenashev conjectured that the Hilbert function of Bbbk[x₁,…,xₙ]/(x₁²,…,xₙ²,ℓ₁²,ℓ₂²), for general linear forms ℓ₁,ℓ₂, is a lattice-path count a(n,d); Booth, Singh and Vraciu translated the conjecture at (n,d) into the assertion dimkerφ_(n+1,d)=Δ(n,d), where φ_(m,d): A_(d-1) → A_d is multiplication by a general linear form and Δ is the excess of a over its binomial approximation, and they exhibited exactly one element of that kernel. Boij and Lundqvist have since proved the conjecture for all n and all degrees, by a Hilbert-series computation that produces no element of the kernel at all. The dimension is therefore known; the elements are not. We construct them. For every subset L₀ ⊆ [m] of size k and every integer s ≥ 0 we write down a pair Rel(L₀, s) ∈ E_(d-1), Cel(L₀, s) ∈ E_(d-2) with L · Rel(L₀, s)=e² · Cel(L₀, s) identically in ℤ[a₁,…,aₘ], so that Rel(L₀, s) lies in kerφ_(m,d) for a general linear form L=Σ aᵢxᵢ; the member k=0, s=m-2d+2 is the element of Booth, Singh and Vraciu, and the range of validity is sharp at its lower end. On the diagonal line m=3d-3-k, where the parameter s=d-1-k is forced, the resulting C(m, k) elements span the whole of kerφ_(m,d); we prove this at eight parameter sets on the lines k=0,1,2,3, and in particular the m elements attached to the singletons span the n-dimensional kernel on the line n=3d-5, for d=3,4,5,6 — the case Booth, Singh and Vraciu named as the next one. The identities are verified as polynomial identities over ℤ[a], and the independence at one integer specialisation — an asymmetry forced by the problem, since a rank at a point bounds the generic rank only from below. We record why no single numerical specialisation can decide the conjecture in either direction, and one transcription correction to the literature. Every computation reported below is machine-checked in Lean 4; the two places where such a computation is combined with a published upper bound to give an equality are marked as such.
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Source snapshot 2026-09-07 03:53 UTC
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Claim ledger
Stated results
W1known data2026-09-03
The lattice-path count a(n,d) of CLN Conjecture 2, computed from the definition: their printed table rows n = 3,5,7,9, the vanishing of the excess on CLN's Proposition 7 range d <= n/3+1, and the excess 1 on the line n = 3d-4 settled by Booth-Singh-Vraciu
W2known data2026-09-03
Delta (3d-5) d = 3d-5 for d = 3..12 – the assertion T(n,d) = n of Booth-Singh-Vraciu Remark 5.6 (referee 2026-09-03; v3 numbering), recomputed from the path definition, with too-large and too-small controls
W3known2026-09-03
The excess on the next lines: Delta (3d-6) d = C(n,2) - 2, Delta (3d-7) d = C(n,3) - 2n, Delta (3d-8) d = C(n,4) - 2C(n,2) + 1, and the two-term closed form a(n,d) = F(n,d) + F(n,3d-4-n) with F(n,m) = C(n,m)-2C(n,m-2)+C(n,m-4)
W4candidate2026-09-03
An explicit family of m kernel elements: L * Rhatₗ = e² * Chatₗ as a polynomial identity in Z[a₁..aₘ] for every l, verified for (m,d) = (4,3),(5,3),(6,4),(7,4),(8,4),(9,5),(10,5),(11,5),(14,6); sharp – it fails at m = 3d-3 – the k = 1 line of the family of W9
W5candidate2026-09-03
The m elements Rhat₁..Rhatₘ span exactly m-1 = n dimensions in A_(d-1) = E_(d-1)/e² E_(d-3): an (m-1)x(m-1) minor of [<xiᵢ, Rhatₗ>] is nonsingular while the full m x m determinant vanishes, for d = 3,4,5,6
W6known2026-09-03
The n = 3d-5 slice of CLN Conjecture 2 for d = 3,4,5,6, by an independent constructive route: dim ker phi_(n+1,d) = n, hence dim (Pₙ/(x₁²,..,xₙ²,l₁²,l₂²))_d = a(n,d) for (n,d) = (4,3),(7,4),(10,5),(13,6)
This ledger entry is reported in prose and is not bound to a Lean theorem.W7prose2026-09-03
Theorem, uniform in k, s, m and d: for m - 2d + 2 <= s and s + k <= d - 1 [AUTHOR FIX 2026-09-03: the lower inequality has NO k – the ledger wrote m - 2d + 2 <= s (and s + k <= d - 1; the 's + k' lower reading was a slip – referee 2026-09-03), which the family's own negative controls (7,4,2,0), (10,5,2,1), (9,5,3,0) refute; the proof needs |T| = m - d - s <= d - 2; only the lower inequality has negative controls] and every L0 of size k, L * R_(L0,s) = e² * C_(L0,s), so R_(L0,s) lies in ker phi_(m,d); and sumₗ Rₗ = (m-2d+2) times the Booth-Singh-Vraciu element
This ledger entry is reported in prose and is not bound to a Lean theorem.W8measurement2026-09-03
Measurement: Conjecture 2 is not checkable by any single numerical specialisation of l₁, l₂, in either direction; a certificate must be generic on one side and a specialisation on the other
This ledger entry is reported in prose and is not bound to a Lean theorem.W9candidate2026-09-03
The k-index deformation family R_(L0,s): L * Rhat_(L0,s) = e² * Chat_(L0,s) over Z[a] exactly when m - 2d + 2 <= s and s + k <= d - 1 [AUTHOR FIX 2026-09-03: the lower inequality has NO k – the ledger wrote m - 2d + 2 <= s (and s + k <= d - 1; the 's + k' lower reading was a slip – referee 2026-09-03), which the family's own negative controls (7,4,2,0), (10,5,2,1), (9,5,3,0) refute; the proof needs |T| = m - d - s <= d - 2; only the lower inequality has negative controls] (verified for (m,d,k,s) = (9,4,0,3),(7,4,2,1),(10,5,2,2),(9,5,3,1),(11,6,4,1), with three sharpness controls), and on the line m = 3d-3-k the C(m,k) elements span the whole kernel (Lean for k = 0,2,3; C for k = 0..4)
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- > Status note (2026-09-03). The conjecture this family was founded on was proved in full > by Mats Boij and Samuel Lundqvist, *On the Hilbert series of ideals generated by general linear > forms*, arXiv:2608.22823v1, submitted 2026-08-24 — nine days before the dispatch and after the > founding source. Their Theorem thm:d2 gives the Hilbert series of R_(n,n+2,2) for every n, > which is exactly CLN Conjecture 2. What is left, and what this family carries, is the > constructive side: Boij–Lundqvist compute the dimension of ker φ_(n+1,d) without producing a > single element
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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