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.

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。

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