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

Background & Foundational Framework 1.1 The Laguerre–Pólya Class LP A real entire function $f(z)$ belongs to the Laguerre–Pólya class LP if it can be represented in the Hadamard product form: f(z) = C zᵐ e-α z² + β z ∏ₖ₌₁^∞ (1 - z/xₖ) ez / xₖ where: C, β, xₖ ∈ R, xₖ ≠ 0, and m ∈ N ∪ \0\; α ≥ 0; ∑ₖ₌₁^∞ xₖ⁻² < ∞. Equivalently, f ∈ LP if and only if $f(z)$ is the uniform limit on compact subsets of C of real polynomials having only real zeros. 1.2 The Strip Class S(A) and Imaginary Shifts For a bandwidth parameter $A > 0$, the strip class S(A) consists of real entire functions whose zeros lie within the closed horizontal strip: Im(z) ≤ A For any real shift parameter μ ∈ R, we define the shifted symmetrized function: f_μ(x) := f(x + iμ) + f(x - iμ) = 2 Re(f(x + iμ)), x ∈ 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) := (g'(x))² - g(x) g''(x) For entire functions f ∈ LP, the classical Laguerre inequality guarantees that L[f](x) ≥ 0 for all x ∈ R. Applying this to shifted combinations, we define the shifted Laguerre quantity: L_μ[f](x) := (f_μ'(x))² - f_μ(x) f_μ''(x) Conjecture (Csordas & Escassut): Let f ∈ S(A) be a real entire function. If f satisfies strict shifted Laguerre positivity: > L_μ[f](x) > 0 for all x ∈ R and all μ ∈ R \0\ > then f must belong to the Laguerre–Pólya class 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 (α > 0). In the Gaussian-free regime (α = 0), non-real zeros generate localized "debts" in the positivity of L_μ[f](x).