NEW(S)偏元数学:方向的回归——动作残差、两个分量与一般和乐 (Prenary Mathematics: The Return of Direction — Action Residuals, the Two Components, and General Holonomy)

本文从一个被忽略的前提出发,给出一个关于"动作"的最小数学框架,并在其上确立一组可证命题。通常的算术默认"相减可以精确归零"(a − a = 0)。本文不采用该前提,而采用其否定。公理 1(动作留差):任一次动作都留下一个非零残差——op(a) = op₀(a) + ε,且 ε ≠ 0。在此前提下,本文给出三层结构(对象层/残差层/接入)、层间的一个投影、以及动作层上的具体对象——偏差与和乐。主要结果:(1)偏差不是端点差的差(非上边界性);(2)纯旋转情形下,闭路和乐正比于步数(Hol = N·(π/2)),与步长成反比;(3)复缩放的显式刻画;(4)"量"侧的偏好 Pref(δ₀) = 1 − 2/δ₀(候选底层定义);(5)一般和乐 Hol := ∏λₙ 具有分离性 ∏λₙ = (∏ρₙ)·e^{i·Σφₙ};(6)退化关系——经典量在退化中被回收,不被破坏。本文同时明标其边界:不含"接入"运算规则、不含可检验物理预测、不含嵌入映射。本文尚未得到独立实验验证。关键词:动作残差;残差上界;层分离;投影;覆盖空间;和乐;复缩放;量的偏好;退化;可证伪性;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 starts from a neglected premise and gives a minimal mathematical framework about "action", on which a set of provable propositions is established. Ordinary arithmetic assumes that subtraction returns exactly to zero (a − a = 0). This paper does not adopt that premise, but its negation. Axiom 1 (action leaves a residual): every action leaves a non-zero residual — op(a) = op₀(a) + ε, with ε ≠ 0. Under this premise the paper gives a three-layer structure (object layer / residual layer / incorporation), a projection between the layers, and concrete objects on the action layer — the deviation and the holonomy. Main results: (1) the deviation is not the difference of endpoint differences (non-boundary-ness); (2) in the pure-rotation case the holonomy is proportional to the step count (Hol = N·(π/2)), hence inversely proportional to the step length; (3) an explicit characterisation of complex scaling; (4) the preference on the "quantity" side, Pref(δ₀) = 1 − 2/δ₀ (a candidate underlying definition); (5) the general holonomy Hol := ∏λₙ has the separability ∏λₙ = (∏ρₙ)·e^{i·Σφₙ}; (6) a degeneration relation — the classical quantity is recovered, not destroyed. The paper marks its boundaries explicitly: no operation rules for "incorporation", no testable physical predictions, no embedding map. This work has not yet been independently verified by experiment. Keywords: Action Residual, Residual Upper Bound, Layer Separation, Projection, Covering Space, Holonomy, Complex Scaling, Preference of Quantity, Degeneration, Falsifiability, Lean 4 Formal Verification, Chen Pianzhen, Chensong_AI_LateNightLab, Huaxia Sizhe Pianyuan Zhu, PGI Jiaolong. — Published by Lao Chen & AI's Late Night Lab. Please accept with a smile.