V142_1 — Partial Fractions and Separation Bounds for the M = 1 Low-Strip Gram Kernel

Description: This paper provides an explicit structural analysis of the weighted Gram kernel that arises in the enlarged low‑frequency strip at the natural M = 1 scale. Previous work treated the off‑diagonal Gram form abstractly through Bessel conditions; here, the kernel is studied directly.

The central contribution is an exact partial‑fraction reduction of the two‑packet Gram kernel into one‑packet transforms and a separation denominator. A uniform bound for the one‑packet transform is established, showing that off‑diagonal interactions weaken as the separation between ordinates increases. This yields the first direct separation estimate for the weighted Gram kernel: far‑pair interactions are explicitly damped, while close‑pair clustering remains the delicate regime.

The paper emphasizes that the separation estimate does not by itself prove a Schur or Bessel bound. Near pairs require stronger control, and far pairs still demand pair‑counting, cancellation, or phase information. Thus the off‑diagonal obstruction is reframed as a separation‑weighted pair‑summation problem.

No proof of the Riemann Hypothesis is claimed. The contribution is diagnostic: it isolates the off‑diagonal obstruction, reduces it to a separation kernel, and highlights the need for phase analysis or pair‑correlation input to resolve the remaining difficulty.