bibkey: lee2026multivariateindependence authors: Joonkyung Lee; Jaehyeon Seo year: 2026 title: “Lower bounds for multivariate independence polynomials and their generalisations” doi: 10.48550/arXiv.2602.02450 url: https://arxiv.org/abs/2602.02450v2 claim: “Section 5 proposes the occupancy-fraction strengthening (5.2), equivalent to sum_v p_v(lambda) >= sum_v lambda_v/(1+(d_v+1)lambda_v) for all nonnegative vertex fugacities.” strata_touched:

• D5/S3/StatisticalMechanics/HardCore/MultivariateOccupancyRefutation license: citation-only triage: anchor

Lee and Seo, multivariate independence polynomials

Section 5, page 16, under “Occupancy fractions”, states:

Having seen the Davies–Kang conjecture, it seems plausible to look for a strengthening of Theorem 1.1 in terms of occupancy fractions.

The vertex occupancy p_v(λ) := λ_v ∂/∂λ_v log Z_G(λ) is the probability that v belongs to a hard-core independent set. The paper defines α_G(t; λ) = (1/|V(G)|) Σ_{v∈V(G)} p_v(tλ) and proposes

$${\alpha }_G(t;\lambda )\ge \frac{1}{|V(G)|}\sum _{v\in V(G)}{\alpha }_{K_{d_v+1}}(t{\lambda }_v),t\in {\mathbb{R}}_{\ge 0},\lambda \in ({\mathbb{R}}_{\ge 0}{)}^{V(G)}.\text{\hspace{1em}(5.2)}$$

Here α_{K_{d+1}}(x) = x/(1+(d+1)x) by (5.1). The subsequent sentence reads:

Moreover, (5.2) is equivalent to showing that for all λ ∈ (R_{≥0})^{V(G)},

$$\sum _{v\in V(G)}{p}_v(\lambda )\ge \sum _{v\in V(G)}{\alpha }_{K_{d_v+1}}({\lambda }_v).$$

The proposed statement fails already at t = 1 on K_{1,2} with fugacities (15,2,2): the sum of occupancies is 9/8, smaller than 15/46 + 2/5 + 2/5 = 259/230. All entries are strictly positive. This disproves the proposed strengthening, and does not contradict the partition-function lower bound of Theorem 1.1: integrating a proposed stronger pointwise bound is a sufficient route to that theorem.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2602.02450
• URL: https://arxiv.org/abs/2602.02450v2
• Version 2, Section 5, page 16, “Occupancy fractions”, equations (5.1), (5.2), and the equivalent vertex-marginal inequality.