Smooth Integer Encoding via Integral Balance

ORCID: 0000-0002-5891-8119 April 28, 2025 Abstract We introduce a novel method for encoding integers using smooth real-valued functions whose integral properties implicitly reflect discrete quantities. In contrast to classical representations, where the integer appears as an explicit parameter, our approach encodes the number N N through the cumulative balance of a smooth function f N(t), constructed from localized Gaussian bumps with alternating and decaying coefficients. The total integral I ( N) converges to zero as N, and the integer can be recovered as the minimal point of near-cancellation. This method enables continuous and differentiable representations of discrete states, supports recovery through spline-based or analytical inversion, and extends naturally to multidimensional tuples ( N 1, N 2, .. . We analyze the structure and convergence of the encoding series, demonstrate numerical construction of the integral map I (N), and develop procedures for integer recovery via numerical inversion. The resulting framework opens a path toward embedding discrete logic within continuous optimization pipelines, machine learning architectures, and smooth symbolic computation. Numerical Analysis 1 Introduction Representing discrete quantities such as integers within continuous mathematical frameworks is a central challenge in optimization, numerical analysis, and machine learning. Traditional symbolic representations and modern soft relaxation techniques both face fundamental limitations: the former lack differentiability, while the latter introduce approximation errors and auxiliary complexities. In this work, we propose a novel method for encoding integers through smooth real-valued functions whose integral properties implicitly reflect discrete quantities.