用光滑函数积分实现整数的连续可微编码，支持优化与学习。
Smooth Integer Encoding via Integral Balance
- 通过带符号衰减高斯波峰的积分和隐式编码整数
- 积分值随整数增大趋近零，最小点对应原数
- 适合嵌入神经网络等连续优化系统
我们提出一种新方法，用光滑实值函数的积分性质隐式编码自然数。不同于传统显式表示，该方法通过构造带有交替衰减系数的局部高斯波峰函数 $f_N(t)$，将整数 $N$ 编码为其累积平衡的积分 $I(N)$。当 $N$ 趋于无穷时，$I(N)$ 收敛于零，整数可通过近似抵消的最小点恢复。该方法实现离散状态的连续可微表示，支持基于样条或解析反演的恢复，且可自然扩展至多维元组 $(N_1, N_2, ext{...})$。我们分析了编码级数的结构与收敛性，展示了 $I(N)$ 的数值构造，并提出了数值反演的恢复流程。该框架为在连续优化、机器学习架构和光滑符号计算中嵌入离散逻辑开辟了新路径。
原文摘要 · Abstract (English)
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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