Collective Screening and Real-Mesh Obstructions in the Shifted Laguerre Converse

Abstract

1. Background & Foundational Framework 1.1 The Laguerre–Pólya Class $\mathcal{LP}$ A real entire function $f(z)$ belongs to the Laguerre–Pólya class $\mathcal{LP}$ if it can be represented in the Hadamard product form: $$f(z) = C z^m e^{-\alpha z^2 + \beta z} \prod_{k=1}^\infty \left(1 - \frac{z}{x_k}\right) e^{z / x_k}$$ where: $C, \beta, x_k \in \mathbb{R}$, $x_k \neq 0$, and $m \in \mathbb{N} \cup \{0\}$; $\alpha \ge 0$; $\sum_{k=1}^\infty x_k^{-2} < \infty$. Equivalently, $f \in \mathcal{LP}$ if and only if $f(z)$ is the uniform limit on compact subsets of $\mathbb{C}$ of real polynomials having only real zeros. 1.2 The Strip Class $\mathcal{S}(A)$ and Imaginary Shifts For a bandwidth parameter $A > 0$, the strip class $\mathcal{S}(A)$ consists of real entire functions whose zeros lie within the closed horizontal strip: $$\vert{}\operatorname{Im}(z)\vert{} \le A$$ For any real shift parameter $\mu \in \mathbb{R}$, we define the shifted symmetrized function: $$f_\mu(x) := f(x + i\mu) + f(x - i\mu) = 2 \operatorname{Re}\bigl(f(x + i\mu)\bigr), \quad x \in \mathbb{R}$$ 1.3 The Shifted Laguerre Inequality and the Conjecture For a twice-differentiable real-valued function $g(x)$, the classical Laguerre expression is defined by: $$L[g](x) := \bigl(g'(x)\bigr)^2 - g(x) g''(x)$$ For entire functions $f \in \mathcal{LP}$, the classical Laguerre inequality guarantees that $L[f](x) \ge 0$ for all $x \in \mathbb{R}$. Applying this to shifted combinations, we define the shifted Laguerre quantity: $$L_\mu[f](x) := \bigl(f_\mu'(x)\bigr)^2 - f_\mu(x) f_\mu''(x)$$ Conjecture (Csordas & Escassut): Let $f \in \mathcal{S}(A)$ be a real entire function. If $f$ satisfies strict shifted Laguerre positivity: $$> L_\mu[f](x) > 0 \quad \text{for all } x \in \mathbb{R} \text{ and all } \mu \in \mathbb{R} \setminus \{0\} >$$ then $f$ must belong to the Laguerre–Pólya class $\mathcal{LP}$ (i.e., all zeros of $f$ must be real, or $A = 0$). 2. Obstruction Theory & The Screening Mechanism When non-real zeros are present, the logarithmic derivative of $f(z)$ generates electrostatic interactions along the real axis. Prior work established obstructions in the positive-Gaussian regime ($\alpha > 0$). In the Gaussian-free regime ($\alpha = 0$), non-real zeros generate localized "debts" in the positivity of $L_\mu[f](x)$.