Two-Point Boundary Layers in Normalized Gaussian Averages on Discrete Supports

Consider a countable discrete support on the real line whose points carry bounded labels. Gaussian weights turn these data into a normalized field, but the contribution of points outside a chosen adjacent pair can obscure its local transition. We separate the field exactly into the normalized response of that pair and the weighted average of the remaining labels. The pair response is logistic in the Gaussian log-ratio, a binary softmax structure; an exact error identity expresses the effect of the remaining points through their share of total weight. A cellwise geometric estimate bounds that share using the distances from external support points to the pair. For the support {log n : n ≥ 2}, we identify the local scale ν = (m+2)h and obtain an explicit error bound uniform over the cell index m. Consequently, h = o(1/m) suffices for uniform recovery of the two-point profile. Prime-indicator labels provide a local arithmetic case study, while alternative labels and an irregular support show which parts of the analysis do not depend on primality. Numerical illustrations and kernel-regression examples clarify the scope and limitations of the bounds.