NEW(S)偏元数学地基定稿:动作留差公理与对象层/动作层分离(Prenary Mathematics Foundation: The Action-Residual Axiom and Object–Action Layer Separation)

本文确立偏元数学新线的最小数学地基。核心主张:偏元数学不改变经典数学的对象层,只增加一条动作留差公理——任何动作都不完美闭合,实际结果与经典结果之间始终留下一个不可消除的正残差 ε ∈ (0, δ₀),δ₀ 为不可达到的上界。当 δ₀ = 0 时动作层为空,偏元数学完全退化为经典数学(赋值退化)。对象层 = 经典复数域(Prenary = ℂ,不携带方向、状态或隐藏属性);动作层独立存在。本文给出三份配套 Lean 4 形式化验证文件(Day22-01/02/03),覆盖:动作残差的存在性、正性、开区间与不可达上界;动作函数的非平凡性与动作不可清零;退化链完整闭合与经典恢复。三份均通过 Lean 4 内核验证(No goals)与 Comparator 二次验证,并完成两轮复现(2026-09-09 / 2026-09-10)。本文是一篇独立成篇的尝试性数学工作。ε 是作为公理引入的,不是从经典数学推导出来的结论。本文报告一个边界清楚、可复现、可退化的公理框架,并给出明确的可证伪条件,欢迎独立复核与反驳。本文尚未得到独立实验验证。配套仓库:prenary-math-lean-s2-action-foundation,DOI 10.5281/zenodo.22719535。——老陈与AI的深夜实验室 发布 请笑纳—— This paper establishes a minimal mathematical foundation for the new line of Prenary Mathematics. Core claim: Prenary Mathematics does not change the object layer of classical mathematics; it adds one axiom—the action residual: no action closes perfectly, and between the actual result and the classical result there always remains an irreducible positive residual ε ∈ (0, δ₀), with δ₀ an unreachable upper bound. When δ₀ = 0 the action layer is empty and Prenary Mathematics fully degenerates into classical mathematics (assignment degeneration).The object layer is the classical complex field (Prenary = ℂ, carrying no direction, state, or hidden attribute). The action layer exists independently. This paper presents three companion Lean 4 verification files (Day22-01/02/03), verifying: the existence, positivity, open-interval and unreachable-bound of the action residual; the non-triviality and non-return of the action function; and the complete closure of the degeneration chain and classical recovery. All three pass Lean 4 kernel verification (No goals) and Comparator secondary verification, with two independent runs (2026-09-09 and 2026-09-10).This paper is a self-contained, tentative mathematical work. ε is introduced as an axiom, not derived from classical mathematics. It reports a well-bounded, reproducible, degenerable axiomatic framework and gives explicit falsifiability conditions, inviting independent scrutiny and refutation. It has not yet been validated by independent experiment.Companion repository: prenary-math-lean-s2-action-foundation, DOI 10.5281/zenodo.22719535.— Published by Lao Chen & AI's Late Night Lab. Please accept with a smile.