Smooth Integer Encoding via Integral Balance

TL;DR

This paper presents a new smooth integer encoding method using integral properties of constructed functions, enabling continuous representations of discrete numbers for applications in optimization and machine learning.

Contribution

The paper introduces a novel smooth encoding of integers via integral balance, allowing continuous and differentiable representations of discrete states with inversion procedures.

Findings

Encoding converges as N increases

Integer recovery via numerical inversion is feasible

Supports multidimensional tuple encoding

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 in the set of natural numbers 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 tends to infinity, 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 (N1, N2, ...). 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.