基于双属性微分单元的代数演算框架及其在奇点赋值中的应用

An Algebraic Calculus Framework Based on Dual-Attribute Differential Units and Its Application in Singularity Assignment

摘要: 本文提供了一种基于双属性微分单元的代数微积分框架。核心创新在于将微分单元$dx$ 定义为同时具备零值属性($\left[dx\right]=0$ )与非零征属性($\left\{dx\right\}\ne 0$ )的实体，其中征属性决定了其在代数运算中的结构性行为(而非单纯数值为零)。在该体系下，经典极限过程为微分单元的约分与归零。除罗尔定理作为光滑闭区间上的归纳锚点外，拉格朗日中值定理、柯西中值定理、洛必达法则及泰勒展开均可经由传统代数路径推导，且结果与经典微积分同构。作为对奇点处理的扩展，本文引入对称导数定义，并将其应用于魏尔斯特拉斯函数等经典病态案例。该框架能在整数格点处给出确定的代数赋值(如$W^{\prime }\left(k\right)=0$ )，同时在非对称点保留经典分析的发散本性，从而对连续统与奇点进行代数描述。

Abstract: This paper presents an algebraic calculus framework based on dual-attribute differential units. The core innovation lies in defining the differential unit $dx$ as an entity possessing both a zero-value attribute ($\left[dx\right]=0$ ) and a non-zero characteristic attribute ($\left\{dx\right\}\ne 0$ ), where the characteristic attribute determines its structural behavior in algebraic operations (rather than being merely numerically zero). Within this framework, the classical limit process corresponds to the reduction and nullification of differential units. Apart from Rolle’s Theorem serving as an inductive anchor for smooth closed intervals, Lagrange’s Mean Value Theorem, Cauchy’s Mean Value Theorem, L’Hôpital’s Rule, and Taylor expansion can all be derived via traditional algebraic paths, yielding results that are isomorphic to classical calculus. As an extension for handling singularities, this paper introduces the symmetric derivative and applies it to classic pathological cases such as the Weierstrass function. This framework provides definite algebraic assignments at integer grid points (e.g., $W^{\prime }\left(k\right)=0$ ) while preserving the divergent nature of classical analysis at asymmetric points, thereby offering an algebraic description of the continuum and singularities.