Withdrawn by the author. This document was produced in an exploratory project with AI language models (2025–2026). An independent check in September 2026 found that it contains mathematical errors and novelty claims that cannot be upheld. Its correct mathematical content is either standard theory or is covered, with complete proofs, in:C. Hartmann, Second-order generators with quadratic Leibniz defect on M_n: a self-contained study of ½[H,[H,·]] + [K,·], Zenodo (2026), DOI: [https://doi.org/10.5281/zenodo.22967633].Please cite that text instead. This version is kept for the record only.
Withdrawn by the author. This document was produced in an exploratory project with AI language models (2025–2026). An independent check in September 2026 found that it contains mathematical errors and novelty claims that cannot be upheld. Its correct mathematical content is either standard theory or is covered, with complete proofs, in:C. Hartmann, Second-order generators with quadratic Leibniz defect on M_n: a self-contained study of ½[H,[H,·]] + [K,·], Zenodo (2026), DOI: [https://doi.org/10.5281/zenodo.22967633].Please cite that text instead. This version is kept for the record only.
⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This paper introduces the Order-Projection Principle (OPP), a minimal andmodel-independent framework describing when order-dependent observables becomevisible in systems combining noncommutative operations and irreversibleprojection. This record is provided for archival and priority purposes. This manuscript was prepared by the author, with AI tools used for language drafting and organization. The scientific content and conclusions are the author’s own.
> ⚠️ **【错误声明 / Error Statement】**> 本条目为作者借助 AI 辅助整理并公开的第一版本(v1.0,2026-08-11),**文章存在严重错误**,现予以保留作为思想演进的原始记录。>> **核心错误:** 本版本错误地将极限构造
limn→∞(1+iθ/n)n 直接定义为“模长恒为1的纯旋转算子”。作者当时未意识到,该极限过程中的**每一步**均存在**缩放泄漏(Scaling Leakage)**,其模长偏差为
eθ2/2n−1=0(当
n 为有限值时)。这是经典数学中“操作过程”向“代数结果”妥协的产物。>> **修正情况:** 此严重错误已由作者于次日(2026-08-12)通过独立物理直觉发现并彻底修正。**请勿以本版本为准**,请务必查阅并引用后续修正版本(v2.0)。>> **保留原因:** 保留此错误版本,旨在完整记录从“极限陷阱”到“操作优先公理”的认知跃迁过程,印证“我怀疑一切,甚至怀疑自己的怀疑”这一方法论的有效性。>> --->> **English Statement:** > **This entry is v1.0 (initial upload: 2026-08-11). AI assistance was utilized in the organization and formatting of this manuscript. It contains a critical error and is preserved as an unaltered historical record of the ideation process.** >>> **Core Error:** This version erroneously defines the limit construction
limn→∞(1+iθ/n)n as a "pure rotation operator of unit modulus." It failed to recognize that **each step** of this limiting process exhibits **scaling leakage**, with a modulus deviation of
eθ2/2n−1=0 for any finite
n. This represents a compromise where the "operational process" is subordinated to the "algebraic result.">> **Correction:** This critical error was independently identified and definitively corrected by the author on the following day (2026-08-12) via physical intuition. **Do not cite or rely on this version.** Please consult and cite the revised version (v2.0).>> **Reason for Preservation:** This v1.0 entry is retained to document the complete cognitive leap from the "pitfall of limits" to the "axioms of operation-first priority," validating the methodology of "I doubt everything, even my own doubts." 本文提出一个基于“操作优先”视角的替代性数学语言框架:将经典欧拉公式中的旋转与缩放属性显式解耦,定义模长恒为1的旋转算子
Rotθ(1)。本文不改变标准量子力学的任何可观测预言,仅为理解量子相位表示、纠缠态归一化、量子纠错开销等问题提供新的重参数化方案与概念视角。本文为预印本,未经同行评审,所有结论待后续理论验证与实验检验。 ### 核心内容 1. **修正定义**:通过极限构造
Rotθ(1)=limn→∞(1+iθ/n)n=cosθ+isinθ,显式分离指数运算中的缩放因子与旋转分量,避免经典记法中底数
e的缩放属性被隐性隐藏。 2. **量子应用框架**: - 单粒子演化层面,修正记法引入的缩放因子可被幺正归一化完全吸收,不产生可观测效应; - 多体纠缠态层面,修正记法要求同步调整归一化系数,显式暴露经典记法中被模方运算掩盖的缩放因子累积效应,其工程学注记提示该形式可作为量子信道非平衡损耗的可监测指标; - 框架为量子纠错归一化开销、纠缠蒸馏效率上限、多体原子钟相干时间优化等问题提供新的概念分析视角。 3. **兼容性说明**:本框架不引入新的物理常数或相互作用,仅对现有数学描述进行重参数化,所有可观测量计算结果与标准量子力学体系完全一致。 ### 关联条目 本文为修正欧拉公式系列研究的第二部分,关联基础理论研究条目:*修正欧拉公式:一个旋转-缩放解耦的操作优先框架*(doi:10.5281/zenodo.21736162)。 ### 身份与合规声明 **本文作者为独立业余研究者,未受任何科研机构、企业项目资助,本研究为个人独立学术探索成果,不涉及任何机构职务成果归属争议,与任何任职/关联单位无利益冲突。** 本文档由AI辅助完成文字表达与格式排版,不涉及国家秘密,仅为公开学术探讨,欢迎学界独立验证与合理引用。 ### 版权与引用 本文采用CC BY 4.0协议开放授权,引用请注明作者与Zenodo DOI。 This preprint proposes an alternative mathematical framework based on the *operation-first* principle: explicitly decoupling the rotational and scaling components inherent in Euler's formula, and defining a unit-modulus rotation operator
Rotθ(1). This work does not modify any observable predictions of standard quantum mechanics, but provides a novel reparameterization scheme and conceptual perspective for understanding quantum phase representation, entanglement state normalization, and quantum error correction overhead. ### Core Contributions 1. **Modified Definition**: The operator
Rotθ(1)=limn→∞(1+iθ/n)n=cosθ+isinθ is constructed via limit definition, explicitly separating the scaling factor from the rotational component in exponential operations, avoiding the implicit hiding of the base
e's scaling property in classical notation. 2. **Application Framework for Quantum Mechanics**: - At the single-particle evolution level, the scaling factor introduced by the modified notation is fully absorbed by unitary normalization, producing no observable effects; - At the multi-particle entanglement level, the modified notation requires synchronous adjustment of normalization coefficients, explicitly exposing the cumulative scaling effect masked by modulus squaring in classical notation. An engineering note suggests this form can serve as a monitorable metric for unbalanced loss in quantum channels; - The framework provides a new conceptual lens for analyzing normalization overhead in quantum error correction, efficiency limits of entanglement distillation, and coherence time optimization in many-body atomic clocks. 3. **Compatibility Statement**: This framework introduces no new physical constants or interactions, only reparameterizing existing mathematical descriptions. All observable calculation results are fully consistent with standard quantum mechanics. ### Related Work This work is the second part of the *Modified Euler's Formula* series, linked to the foundational theoretical entry: *Modified Euler's Formula: An Operation-First Framework for Decoupling Rotation and Scaling* (doi:10.5281/zenodo.21736162). ### Affiliation & Compliance Declaration **The author is an independent amateur researcher with no funding from any academic institution or corporate project. This work is an independent academic exploration by the author, with no conflicts of interest or affiliation with any institution, and no disputes over institutional intellectual property.** This document was prepared with AI assistance for text polishing and formatting. It contains no state secrets, is released solely for open academic discussion, and welcomes independent verification and citation by the research community. ### License & Citation This work is licensed under CC BY 4.0. Please cite the author and the Zenodo DOI when referencing this work.