Smooth Integer Encoding via Integral Balance

• URL: http://arxiv.org/abs/2505.02259v1
• Date: Mon, 28 Apr 2025 20:23:53 GMT
• Title: Smooth Integer Encoding via Integral Balance
• Abstract summary: We introduce a novel method for encoding using smooth real-valued functions.<n>Our approach encodes the number N in the set of natural numbers through the cumulative balance of a smooth function f_N(t)<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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