Finite Laguerre Witnesses and Zero‑Crowding Constraints for the Riemann Xi‑Function

Abstract This paper investigates coefficient‑depth problems in products of the form H(u)=(1−u)2E(u)P(u)F(u), where P has nonnegative coefficients, F lies in a ratio‑constrained cone, and E is a product of quadratic factors. Without interior quadratics, a sharp baseline theorem shows that the first negative coefficient occurs at most at index deg⁡P+o+1. We prove that no uniform additive correction depending only on the number of interior quadratics can extend this bound.

Exact rational counterexamples disprove natural finite additive laws already for two distinct quadratics. More substantially, we construct, for every sufficiently large integer N, explicit polynomials producing unbounded additive excess: with two interior factors the first negative coefficient occurs beyond deg⁡P+o+Θ(o). The mechanism is governed by a Gaussian boundary layer. Under Poisson scaling, the discrete coefficient operator converges to the sixth‑order operator L=A2(A2+4)2, with A=Ds−s. We exhibit an explicit monotone profile solving a free‑boundary problem, certify its signs by exact rational interval arithmetic, and transfer positivity back to the discrete coefficients with explicit constants.

We also analyze the associated near‑collision zero geometry and prove that minimal‑displacement selection does not remove the limiting obstruction. A multi‑ordinate boundary‑cluster construction shows that local coefficient tests and local zero counts cannot by themselves exclude such configurations.

These results neither prove nor disprove the Riemann Hypothesis. They isolate a precise obstruction to a broad class of coefficient‑based approaches, demonstrating that additive depth laws fail in the presence of Gaussian boundary layers.

Keywords Coefficient positivity; Laguerre–Pólya class; ratio‑constrained sequences; Gaussian boundary layer; exact rational certificate; Riemann Hypothesis