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

Abstract

Let $f$ be a real entire function in the strip class $\mathcal{S}(A)$, and for $\mu \in \mathbb{R}$ put $f_\mu(x) = f(x+i\mu) + f(x-i\mu)$. Csordas and Escassut asked whether strict shifted-Laguerre positivity $L[f_\mu](x) > 0$ for every real $x$ and every nonzero $\mu$ forces $f$ to belong to the Laguerre--Polya class. We prove two complementary partial converses. First, the conjecture holds for the full subcase in which the canonical Gaussian coefficient is positive. Second, in the Gaussian-free case, every finite-density direction supporting a nonreal directed escape satisfies the strict quantitative bound $A\overline{D}_\sigma > 1.0138$, with Cartwright and height-sensitive consequences. The paper also develops boundary rigidity for $H = \operatorname{Im}(f'/f)$, graph-free finite-amplitude phase locking, Gaussian-free minimal-type growth, critical-point confinement and simplicity, Pruefer monotonicity, vertical modulus-gap averaging, an exact periodic nonreal-tail obstruction, and a smooth regularization of the boundary set through the zeros of $f$. Revision 1.3 adds an endpoint theorem at the same boundary-rigidity level: on any already-existing right-unbounded, zero-free, regular lower boundary end with $H = 0$ and $H_y > 0$, the boundary amplitude cannot tend to infinity. The same calculation gives $\vert{}\cos\theta\vert{} = O(\vert{}f\vert{})$ on a zero-amplitude end. These endpoint results are conditional on existence of the regular lower boundary end and do not import the later activated-selector or corridor-construction machinery. The full Gaussian-free converse remains open; in particular, a general global separator/end theorem, final finite/zero-amplitude corridor contradiction, and the infinite directional-density regime are not settled here.