NEW(S) 偏元数学残差的复数升域与径向/角向分解(Radial–Angular Decomposition of the Prenary Residual in the Complex Domain)

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22832132
Primary Topic
Model Reduction and Neural Networks
Type
preprint
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preprint

NEW(S) 偏元数学残差的复数升域与径向/角向分解(Radial–Angular Decomposition of the Prenary Residual in the Complex Domain)

Song Chen
Zenodo (CERN European Organization for Nuclear Research)
Model Reduction and Neural Networks
preprint

NEW(S) 偏元数学残差的复数升域与径向/角向分解(Radial–Angular Decomposition of the Prenary Residual in the Complex Domain)

Song Chen
preprint en

Abstract

本文在偏元数学的最小地基上做两件事:把动作残差从实数域放到复数域,并给出累积过程的径向/角向分解。 须先说明位置:本文的分解式(定理 1)是复数加法在极坐标下的标准展开,是一个标准恒等式,不是本文提出的新定理。本文的贡献在于:把"残差带方向"这一前提,落在一个可验证、可复现、可退化的形式上,并标出它在体系中承上启下的位置。本文不主张任何物理量的数值对应。 在"动作 = 累积相加"的设定下,精确分解为 r'² = r² + 2rρcosδ + ρ²;当 ρ ≪ r 且 δ 远离 π 时线性化为 Δr ≈ ρcosδ、Δψ ≈ (ρ/r)sinδ。累积满足 z_{n+1} = z_n + ε_n。数值上:精确式误差在机器精度量级(≈2.2×10⁻¹⁶);角向净累积为 √N 量级(无系统性漂移的随机游走)。四条定理已由 Lean 4 内核验证(No goals)并经 Comparator 二次验证——见配套仓库 DOI 10.5281/zenodo.22815027。 本文是一项独立的、尝试性的数学工作,尚未得到独立实验验证,末尾给出明确的可证伪条件,欢迎独立复核与反驳。 关键词:偏元数学;动作留差;残差;复数域;径向/角向分解;模方展开;累积;可证伪;Lean 4 形式化验证;陈偏贞;老陈与AI的深夜实验室;PGI蛟龙;华夏思哲偏元注 ——老陈与AI的深夜实验室 发布 请笑纳—— This paper does two things on the minimal foundation of Prenary Mathematics: it lifts the action residual from the real field into the complex field, and it gives a radial–angular decomposition of the accumulation process. The decomposition (Theorem 1) is the standard expansion of complex addition in polar coordinates—a standard identity, not a new theorem proposed by this paper. The contribution is to place the premise "the residual carries a direction" onto a verifiable, reproducible, degenerate form, and to mark the load-bearing position it occupies in the framework. No numerical correspondence to any physical quantity is claimed. Under the setting "action = cumulative addition", the exact decomposition is r'² = r² + 2rρcosδ + ρ²; when ρ ≪ r and δ away from π, it linearizes to Δr ≈ ρcosδ and Δψ ≈ (ρ/r)sinδ. Accumulation satisfies z_{n+1} = z_n + ε_n. Numerics: the exact formula holds to machine precision (≈2.2×10⁻¹⁶); the net angular accumulation scales as √N (a random walk with no systematic drift). The four theorems are verified in Lean 4 (kernel No goals + Comparator) — see the companion repository DOI 10.5281/zenodo.22815027. This is an independent, tentative mathematical work; it has not been independently verified by experiment. Explicit falsifiability conditions are given. Independent scrutiny and refutation are welcome. Keywords: Prenary Mathematics, Action Residual, Residual, Complex Field, Radial–Angular Decomposition, Modulus-Square Expansion, Accumulation, Falsifiability, Lean 4 Formal Verification, Chen Pianzhen, Chensong_AI_LateNightLab, Huaxia Sizhe Pianyuan Zhu — Published by Lao Chen & AI's Late Night Lab. Please accept with a smile.

Zenodo (CERN European Organization for Nuclear Research)
Model Reduction and Neural Networks
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