Smooth Integer Encoding via Integral Balance
This provides a novel encoding method for embedding discrete logic in continuous systems like optimization pipelines and machine learning architectures, though it appears incremental as an alternative representation rather than a fundamental breakthrough.
The paper tackles the problem of encoding integers using smooth real-valued functions, where the integer N is encoded through the cumulative balance of a function f_N(t) constructed from Gaussian bumps, enabling continuous and differentiable representations of discrete states. The result is a framework that supports integer recovery via numerical inversion and extends to multidimensional tuples, facilitating embedding discrete logic in continuous optimization and machine learning.
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.