The Quantitative Identity of the Remainder: From ρ≠∅ to Euler's Formula / 余项的定量身份:从ρ≠∅到欧拉公式

Description (English)

ZFCρ Series, Paper II. This paper asks where the quantitative identity of the formalization remainder (ρ) comes from. π is presented as the first complete instance: its existence follows from the ρ-proposition (Paper I), but its value is locked by the Fourier self-dual fixed-point condition at the next structural layer. The paper advances an interpretive re-reading of Euler's formula e^{iπ}+1=0: the exponential map (act, signed by e) binds two remainders (i from algebraic closure, π from harmonic-analytic duality) to produce closure. A structural parallel is drawn with the L₂→L₃ transition using known Turing-degree collapses. A research program is proposed: self-referential generation as a candidate for the unified act driving all layer transitions. Bilingual Chinese–English edition.

Version 2 (July 2026). This version incorporates four errata raised against the paper by SAE Mathematics Paper 5 (DOI: 10.5281/zenodo.21538494), together with one clarification added on its own account. Substantively: (i) remainder 2 of the L₂→L₃ transition is replaced — "the class of Σ₁ truth predicates" cannot carry undefinability, since true Σ₁ sentences form a definable, effectively enumerable set of degree 0′; the remainder is the absence of any unifying member in the family of stratified truth predicates, witnessed by Th(ℕ) of degree 0^(ω); (ii) the Turing-degree equation is unchanged, but its reading is corrected — what coincides at 0′ are two products of the act together with the first non-trivial fragment of the truth hierarchy, not the two remainders and the act; (iii) "the exponential map is the sole source of the relation between i and π" is narrowed to "the canonical binding source within this trajectory", since bridges exist that do not pass through the complex exponential; (iv) two notes are added recording that the ideal of arithmetical degrees has no least upper bound, and that the re-zeroing equation has not been shown to exhibit both remainders as non-deletable inputs. Version 1 is not superseded; its DOI is retained. A full change log appears at the head of the paper.

Description (中文)

ZFCρ系列 论文二。本文追问形式化余项(ρ)的定量身份从何而来。π是第一个完整样例:其存在性由ρ命题(第一篇)保证,其值由高一层的Fourier自对偶不动点条件锁定。本文对欧拉公式e^{iπ}+1=0提出解释性重读:指数映射(行为,以e标记)绑定两个余项(代数闭合的i,调和分析对偶的π)产生闭合。基于已知递归论结果,与L₂→L₃跃迁建立结构平行。提出研究纲领:自指生成作为驱动所有层间跃迁的统一行为候选。中英文双语版。

第二版 (2026 年 7 月). 本版吸收 SAE 数学 Paper 5 (DOI: 10.5281/zenodo.21538494) 对本文提出的四条勘误, 另加一条自行追加的澄清. 实质改动: (i) L₂→L₃ 跃迁的余项二被替换 —— 「Σ₁ 真值谓词类」不能承担不可定义性, 因真 Σ₁ 句集可定义、可有效枚举, 度为 0′; 余项应为分层真谓词族无统一成员, 其见证是度为 0^(ω) 的 Th(ℕ); (ii) 图灵度等式一字未改, 但其解释被更正 —— 在 0′ 处重合的是行为的两个产物与真理分层的第一个非平凡片段, 不是两个余项与行为; (iii)「指数映射是 i 与 π 关系的唯一来源」收窄为「本 trajectory 中的规范绑定来源」, 因存在不经复指数的桥; (iv) 增补两条注, 记录算术度理想没有最小上界, 以及归零方程尚未被证明以两余项为不可删除输入. 第一版不被覆盖, 其 DOI 保留. 完整变更登记见论文开头.

Keywords

ZFCρ, remainder, formalization, Euler's formula, self-referential generation, Turing degrees, exponential map, closure, meta-theory, philosophy of mathematics, SAE framework

Related Identifiers

• Is supplement to: DOI 10.5281/zenodo.18914682 (ZFCρ Paper I)
• References: DOI 10.5281/zenodo.18528813 (SAE Paper 1)
• References: DOI 10.5281/zenodo.18727327 (SAE Paper 3: Complete Framework)
• References: DOI 10.5281/zenodo.18842450 (SAE Methodological Overview)

License

Creative Commons Attribution 4.0 International (CC BY 4.0)

Language

English, Chinese (Mandarin)

Subjects

• Philosophy of mathematics
• Foundations of mathematics
• Meta-theory
• Recursion theory
• Complex analysis