Critical Effective Rank and Sparse Zero Bursts in the Shifted Laguerre Converse

Let f be a real entire function in the standard horizontal-strip class, with zero Gaussian coefficient, and suppose that every nonzero vertical shift satisfies the strict Laguerre inequality. We study vanishing-height nonreal critical points of f, equivalently zeros of the logarithmic derivative $h = f'/f$. At a critical point qₙ = xₙ + i yₙ, we introduce scaled inverse-zero coordinates wj,n = yₙ/zⱼ - xₙ and an effective quadratic rank measuring how many leading coordinates are required to capture the relevant squared inverse-zero mass. The normalized critical blow-up has an exact logarithmic Taylor expansion whose nonlinear coefficients are inverse-zero power sums, while criticality determines the linear coefficient from the higher power sums. Combining this identity with Biró's sharpened Turán power-sum lower bound and a first-order shifted-Laguerre rigidity theorem, we prove that a vanishing-height critical sequence has either a zero at distance comparable with yₙ from its real projection or effective rank tending to infinity. A dyadic-depth collapse theorem then forces a genuinely large annular population, which becomes a horizontal zero interval with both divergent population and divergent local density. Splitting the full zero divisor yields a two-channel alternative: after passage to a subsequence, either the upper-half-plane zero population has divergent local density or the real-zero multiplicity does. Under finite linear upper density in both horizontal directions, the corresponding burst intervals have sublinear width relative to their distance from the origin. The upper-nonreal branch supplies an upstream input to a companion mesoscopic-screening theory, while the real-zero branch remains a separate open sector. No complete proof of the Csordas–Escassut converse is claimed.