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

Abstract This paper develops a Laguerre‑coefficient framework for the Riemann Hypothesis (RH). For each real ordinate t, the normalized product M(a+it)M(a−it) expands into a family of horizontal Laguerre coefficients. We prove that RH is equivalent to the nonnegativity of all such coefficients.

A quantitative finite‑witness theorem is established: if an off‑critical zero exists at a+iy with 0<a<1/2, then some Laguerre coefficient becomes negative for an index n≤Clog⁡(3+∣y∣), with C an effective constant. Thus the all‑order Laguerre criterion reduces, at each height, to finitely many coefficients of logarithmic size.

We further derive the exact Hadamard‑product decomposition of the first Laguerre defect and show how off‑critical zeros require precise compensation. Extending this, an inverse Newton‑moment argument proves that if the first K coefficients remain nonnegative, then the remaining zeros must contribute absolute inverse‑square mass at least K/(ea2). A local zero‑count estimate then forces another zero within distance proportional to a.

These results isolate a residual obstruction: an off‑critical zero can evade finite Laguerre witnesses only through sufficiently strong zero clustering on the scale of its horizontal displacement from the critical line. The Riemann Hypothesis is not proved here. The contribution is a finite‑order reduction, exact defect identities, and a quantified crowding law that clarifies the remaining obstacle.

Keywords Riemann Hypothesis; xi‑function; Laguerre coefficients; finite witness; Hadamard product; Newton identities; zero clustering; analytic number theory