Partial Resolutions of the Shifted Laguerre Converse: Gaussian Drift and Directional Density Barriers

Let f be a real entire function in the strip class S(A), and define f_μ(x) = f(x + iμ) + f(x - iμ). Csordas and Escassut conjectured that the strict shifted Laguerre condition L[f_μ](x) > 0 for every real x and every nonzero μ forces f to belong to the Laguerre–Pólya class LP. We establish several partial results toward this conjecture: Gaussian Case: We prove the conjecture when the Gaussian factor in the canonical product is nontrivial. Directional Density Obstruction: In the Gaussian-free case, we derive a strict directional zero-density obstruction: any finite-density direction carrying a nonreal directed escape must satisfy A · D̄_σ > 1.0138 For Cartwright functions, this implies that τ A ≤ 1.0138π forces f to belong to the Laguerre–Pólya class. Height-Sensitive Refinement: We obtain a refinement in which A is replaced by √A² - η² when the escape has limiting height η. Structural Obstructions: The argument combines the geometry of the level set Re f = 0, shadow-disk coverings, sparse estimates for canonical products, and a real-axis energy identity for the logarithmic derivative. Model Class Counterexamples: Finally, we rule out a broad model class of counterexamples: if, up to finitely many exceptions, the zeros in the upper half-plane form a finite union of horizontal lattices at a common height, then the shifted Laguerre condition cannot hold. This obstruction follows from a Poisson-kernel representation and the Fourier–Bohr spectrum of an almost-periodic boundary field. Together, these results impose explicit geometric and density constraints that any counterexample to the full conjecture would have to evade.