The Leue Modulation Coefficients (LMC): A Smooth Continuum Embedding of Bounded Arithmetic Data

This repository contains a self-contained presentation of a general construction: how to turn any bounded discrete sequence (values in [−1, 1], placed at scattered points in ℝ³) into a smooth field suitable as the coefficient of a uniformly elliptic differential operator. The construction uses a normalized mollification (a Nadaraya–Watson quotient). The normalization is essential: the unnormalized sum is not bounded by 1, since the mollifier peak grows like ε⁻³. The denominator restores the bound |t(x)| ≤ 1 for every choice of scale and placement — the single property on which all of the elliptic analysis depends. From this bound the paper derives uniform ellipticity, a pointwise flux bound, two-sided energy coercivity, and a Lyapunov dissipation estimate. These results use only the boundedness of the coefficient and standard elliptic theory; they hold for any bounded sequence. As a concrete worked example, the construction is applied to the Leue Modulation Coefficients (LMC) — the normalized elliptic-curve trace values tₚ = aₚ/(2√p), which satisfy |tₚ| ≤ 1 by the Hasse bound. The example illustrates the method but does not restrict it; the arithmetic origin of the coefficients is not used in any proof. This Zenodo package includes: the full research paper (PDF), a Python implementation generating the field and conductivity model, a numerical plot comparing the discrete values with the smoothed field, a cover letter and supporting documentation. The contribution is methodological: a clean, reusable recipe for embedding bounded discrete data as controlled PDE coefficients, with the arithmetic example serving as illustration rather than as the source of the analytic content.