Beurling Nyman Geometry and Gram Matrix Structure, Ladder Density and Polynomial Decay via Mellin Smoothing

We study the Beurling Nyman (BN) family $f_{\theta }(x)=\{\theta \mathrm{/}x\}-\theta \{1\mathrm{/}x\}$ in $L^2((0,1])$ through a multiscale ladder parameterisation ${\theta }_{j,k}=2^{-j}{3}^{-k}$ and the associated Gram matrix structure indexed by ladder distance. Using Mellin analysis and a controlled smoothing operator, we establish a rigorous polynomial decay envelope for off-diagonal Gram entries. Specifically, for a Gaussian-type Mellin multiplier we prove that $$\mathrm{\mid }⟨g_{{\theta }_{j,k}},g_{{\theta }_{j^{\mathrm{\prime }},k^{\mathrm{\prime }}}}⟩\mathrm{\mid }{\ll }_m(1+cd((j,k),(j^{\mathrm{\prime }},k^{\mathrm{\prime }})){)}^{-m}$$ for any $m$ in $\mathbb{N}$, where $c=\min \{\log 2,\log 3\}$ and $d$ denotes the ladder distance. As a consequence we obtain block-compressibility of Gram rows for $m>2$. These results provide a rigorous foundation for sparsity phenomena in the BN system and support constructive spectral approaches. The analysis is unconditional and independent of any hypothesis concerning the zeros of the zeta function.