旋转-缩放正交解耦: 复指数符号的几何重建及其在数值积分中的应用

⚠️ 勘误与纠正声明(版本更新) 本人早几天的欧拉修正存在严重错误。现在统一纠正如下: 修正核心为 a^{ σ+iθ} =a ^σ ⋅(cosθ+i⋅sinθ),其中 σ控制缩放、θ控制旋转,旋转角度与底数无关。 在这个体系下,建议实部和虚部都严格遵循不省略的表达。否则会发生逻辑混乱。 省略书写是造成混乱的原因之一。 实部虚部零值和不操作关系: • σ=0,θ ≠0:伸缩归一的旋转。任何非零模长都归一的旋转。 • σ ≠0,θ=0:只伸缩不旋转(模长 a σ ,相位不变)。 • σ=1,θ=0:不伸缩也不旋转(模长保持底数 a),基准状态。 • σ=0,θ=0:伸缩归一且不旋转(模长归 1,结果为原相位)。 经典 e^{ iθ }代表纯旋转,其实是对 σ=1的省略书写,即纯旋转。但是被误解为 σ=0的省略表达。(2026.08.27) 旋转-缩放正交解耦:复指数符号的几何重建及其在数值积分中的应用 作者身份与研究说明: 本文为独立业余研究者的个人思考记录。作者无高等院校数学专业教育背景,对复分析、数值计算等相关领域无系统学术了解,全文逻辑验证仅通过AI工具辅助完成。研究核心仅聚焦于基础数学定义的逻辑严谨性,不构成正式学术结论,仅供交流参考。 核心内容概述: 本文针对经典复指数记法 a ^σ+iθ 将旋转操作与缩放操作强行捆绑、导致旋转角度依赖底数的几何语义缺陷,提出「底数-指数正交解耦」公理体系: 明确缩放分量由底数与实部独立决定、旋转分量由虚部直接给定且与底数无关,二者几何语义完全正交。基于此公理,自然推导出「显式欧拉步+守恒量投影步」两步保结构数值格式,在简谐振动、浅水方程、Robertson刚性ODE、二维Navier-Stokes涡流、洛伦兹吸引子、三维热传导、非线性薛定谔方程共7类典型场景中验证,可实现长期积分下守恒量漂移的完全消除(如洛伦兹系统从发散修复为有界、NS方程能量精度提升13个数量级)。 外语版本说明: 鉴于作者外语水平有限,本文不提供官方认证的外文译本,非中文读者可自行翻译或参考机器翻译版本,译文准确性由使用者自行负责。 联系方式:amisz@126.com 欢迎各界数学爱好者、专业学者批评斧正。 Rotation–Scaling Orthogonal Decoupling: Geometric Reformulation of Complex Exponential Notation and Its Application to Numerical Integration Author Status & Research Disclaimer: This work constitutes a personal reflective note by an independent amateur researcher. The author has no formal post-secondary education background in mathematics and possesses no systematic academic knowledge of the fields involved (including complex analysis and numerical computation); all logical verification was conducted with the assistance of AI tools. The inquiry focuses exclusively on the logical rigor of foundational mathematical definitions and does not constitute a formally peer-reviewed academic conclusion—it is provided solely for scholarly exchange and reference. Core Contribution Overview: This work identifies the geometric semantic flaw in the classical complex exponential notation a σ+iθ , wherein rotation and scaling operations are forcibly coupled, causing the rotation angle to depend on the chosen base. To resolve this, it proposes a base–exponent orthogonal decoupling axiom framework: scaling components are independently determined by the base and real exponent, while rotation components are defined directly by the imaginary exponent and remain entirely independent of the base, achieving full geometric orthogonality between the two operations. Based on this axiom, a two-step structure-preserving numerical scheme—comprising an explicit Euler step followed by a conserved-quantity projection step—is derived. Validation across seven canonical benchmark scenarios (harmonic oscillator, shallow water equations, stiff Robertson ODE, 2D Navier-Stokes vortex, Lorenz attractor, 3D heat conduction, nonlinear Schrödinger equation) demonstrates complete elimination of long-term conservation drift (e.g., restoring the divergent Lorenz system to bounded behavior, improving Navier-Stokes energy accuracy by 13 orders of magnitude). Language Disclaimer: Due to the author's limited proficiency in foreign languages, no officially certified translated version of this work is provided. Non-Chinese readers may produce or refer to unofficial machine/self-translated versions at their own discretion; the accuracy of such translations is the sole responsibility of the user. Contact: amisz@126.com Critiques, corrections, and scholarly feedback from mathematics enthusiasts and professional researchers are warmly welcomed.