NEW(S)偏元数学:残差与二态的代数归属——动作层—对象层分离、偏元等号与 ℤ 与 ℤ/2 的同源结构(Prenary Mathematics: Algebraic Attribution of Residual and Two-State — Action–Object Layer Separation, the Prenary Equality, and the Common Origin of ℤ and ℤ/2)

本文给出一个结构:同一个式子 a ⊖ a = ε,可以在两层上读——在动作层读作“这个动作的残留”,在对象层读作“这一层的零”。两层各自带一个二态与一份残差;两层之间由一个对角嵌入 D 相连,且 D 是一个同态。本文给出七点结果: 其一,对象层 = ℂ + 二态 + 二态的“型”——二态有“型”(连续型/离散型),且“型”随层而变(数层连续、集合层离散);“离散” = 连续经一次“忘记结构”(= 取商)。 其二,“表里”可用一条同态命题写出:D(x) = (x, 0),且 D(x ⊖ y) = D(x) ⊖ D(y)。 其三,配对的两个分量 =(量, 向);第二分量就是方向;同对象自减得 (ε, 0),异态相减得 (ε, φ−ψ)。ε 的模非负、相位自由。 其四,一条判据与一面镜子:“互藏”判据(凡二选一,先问互藏;若是则同时存在)与 D 与 Robinson transfer 的对照。 其五,在物理学应用中的一例:DAMPE 伽马射线事例样本的残余 r = obs/geo 满足 r ≈ η_obs − η_geo(四带逐带验算对上)——“同一套几何的两种读法相减留差”,与 a ⊖ a = ε 同构;且幅度与相位解耦(corr ≈ 0.06),相位的漂移归因于标架(曝光几何)自身的漂移(corr = 0.983)。 其六,当 ε 不再被当作“可合并的小量”时,两层的迭代给出“经典项 + ε 修正项”的第一条具体陈述——残差往复律:上一步产生的残差,进入下一步后成为对象的成分;下一步产生的残差,已把上一步的贡献折入其中。故过程恒有“一个当前残差”;形式化为接入规则 ε_{n+1} = F(ε_n, x)(F 保开区间 (0, δ₀))。“并列相加”会无界,“折入”自动保界。 其七,“ε 是对象”不等于“ε 是经典对象”(第 14 节):ε 进入下一步后确实住进了对象层,但对象层的“零”是 (ε, φ₀) 而非经典的 0;故经典运算没有“作用到它上”这回事——不是被抵消,是不在同一集合里。 本文为结构层面的推演,尚未得到独立实验验证;不主张与任何外部理论等价,不主张任何物理数值。 老陈与AI的深夜实验室 发布 请笑纳 This paper presents a structure: one and the same formula, a ⊖ a = ε, can be read at two layers — read at the action layer as “the leftover of this action”; read at the object layer as “the zero of this layer”. The two layers each carry a two-state and a residual; the two layers are connected by a diagonal embedding D, and D is a homomorphism. Seven results are given. First, the object layer = ℂ + two-state + the “type” of the two-state — a two-state has a “type” (continuous / discrete), and the “type” varies with the layer; “discrete” = continuous after one act of “forgetting structure” (= taking a quotient). Second, “outside and inside” can be written out by a single homomorphism proposition: D(x) = (x, 0), and D(x ⊖ y) = D(x) ⊖ D(y). Third, the two components of the pair = (quantity, direction); the second component is precisely the direction; subtracting an object from itself gives (ε, 0), subtracting different states gives (ε, φ−ψ). The modulus of ε is non-negative and its phase is free. Fourth, one criterion and one mirror: the “mutual containment” criterion, and a comparison between D and Robinson transfer. Fifth, one example in physical application: for the DAMPE gamma-ray event sample the residual r = obs/geo satisfies r ≈ η_obs − η_geo (verified band by band) — the difference of two readings of one and the same geometry does not vanish; magnitude and phase are decoupled (corr ≈ 0.06), and the phase drift is attributable to the drift of the frame itself (corr = 0.983). Sixth, once ε is no longer treated as a negligible small quantity, the iteration of the two layers gives the first explicit “classical term + ε correction” statement — the residual reciprocation law: the residual produced in the previous step enters the next step as a constituent of the object, and the residual produced in the next step has folded in the previous contribution. Hence there is always exactly one current residual; formalized as an access rule ε_{n+1} = F(ε_n, x) with F preserving the open interval (0, δ₀). Juxtaposed addition diverges; folding-in preserves the bound. Seventh, “ε is an object” is not the same as “ε is a classical object” (Section 14): after entering the next step ε does reside in the object layer, but the zero of that layer is (ε, φ₀), not 0; hence classical operations never “act on it” — it is not cancelled, it is simply not in the same set. This is a structural study. It has not yet received independent experimental verification; it claims no equivalence with any external theory and claims no physical numerical value. Published by Lao Chen & AI Late-Night Lab. Please enjoy.