NEW(S) 偏元数学体系总论:动作留差公理与退化包容的 Lean 4 形式化验证(Prenary Mathematics System Overview: Lean 4 Formal Verification of the Action Residual Axiom and Degeneracy Inclusion)

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-10
DOI
https://doi.org/10.5281/zenodo.22692592
Primary Topic
History and Theory of Mathematics
Type
preprint
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preprint

NEW(S) 偏元数学体系总论:动作留差公理与退化包容的 Lean 4 形式化验证(Prenary Mathematics System Overview: Lean 4 Formal Verification of the Action Residual Axiom and Degeneracy Inclusion)

Song Chen
Zenodo (CERN European Organization for Nuclear Research)
History and Theory of Mathematics
preprint

NEW(S) 偏元数学体系总论:动作留差公理与退化包容的 Lean 4 形式化验证(Prenary Mathematics System Overview: Lean 4 Formal Verification of the Action Residual Axiom and Degeneracy Inclusion)

Song Chen
preprint en

Abstract

本文尝试以 2026-08-12 至 2026-09-06 逐日 Lean 4 形式化验证(Day1-Day21)并经 Lean 4 内核与 Comparator 二次验证的结论为基础,系统整理偏元数学从地基到总集成的完整骨架。核心主张:偏元数学 = 经典数学 + "动作留差"公理——任何动作(运算、变化、测量)都附带产生一个不可消除的差 ε,ε ∈ (0, δ₀],其中 δ₀ 是残差上界。当 δ₀ = 0 时,偏元数学的全部结构退化为经典数学,即偏元数学严格包含经典数学。 本文汇总 Day1-21 全链共 22 个形式化验证仓库、约 200 个机器验证的核心定理与命题。其中 Day21 总集成将 Day1-20 的核心定义与定理统一命名、汇总于单文件,机器验证整体自洽——这是数学侧的"自洽性闭环"。 本文是一篇独立成篇的尝试性数学工作,未声称"已经证明",所有结论欢迎独立复核与形式化验证。本文尚未得到独立实验验证。 附注:本文地基表述(方向偏好二态)的后续修正——方向偏好移出数学侧、动作留差最小地基——见 NEW(S)-004。 ——老陈与AI的深夜实验室 发布 请笑纳—— This paper attempts to systematically organize the complete skeleton of Prenary Mathematics, from its foundations to its total integration, on the basis of the conclusions of day-by-day Lean 4 formal verification (Day1–Day21) from 2026-08-12 to 2026-09-06, further verified twice by the Lean 4 kernel and Comparator. Core claim: Prenary Mathematics = classical mathematics + the "Action Residual" axiom — every action (operation, change, measurement) gives rise to an irreducible difference ε, ε ∈ (0, δ₀], where δ₀ is the Residual Upper Bound. When δ₀ = 0, the entire structure of Prenary Mathematics degenerates into classical mathematics; that is, Prenary Mathematics strictly includes classical mathematics. This paper summarizes the full chain of Day1–21, comprising 22 formal verification repositories and approximately 200 machine-verified core theorems and propositions, covering: the three axioms, foundations, degeneracy, imaginary numbers, number domains, residual, division and square roots, calculus, order, metric and "1", measure spaces, functional analysis, dynamical systems, geometric topology, algebra and number theory, logic and computation, the complex field, category theory, random ε, the nooks and crannies of the edifice of mathematics (26 cuts), the three-layer structure, hierarchical re-infusion and the direction-preference patch, and total integration. The Day21 total integration gathers the core definitions and theorems of Day1–20 into a single file, machine-verified as self-consistent, complete in coverage, and reproducible. This paper is a self-contained, tentative mathematical work; it does not claim to have "already been proved", and all conclusions are open to independent review and formal verification. It has not yet been validated by independent experiment. Note: the subsequent revision of the foundational formulation (Direction Preference Duality) — moving direction preference out of the mathematical side, with the Action Residual minimal foundation — is presented in NEW(S)-004. — Published by Lao Chen & AI's Late Night Lab. Please accept with a smile.

Zenodo (CERN European Organization for Nuclear Research)
Reduced inequalities
History and Theory of Mathematics
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