Ontological Re-reading of Euler's Identity as Topological Self-Inspection — Ontological Correspondences of the Five Constants

Abstract

**[Manifestation layer]** The cosmos is a living organism. **[Verification layer]** Euler’s identity \(e^{i\pi}+1=0\) is the most concise manifestation of the topological self-inspection of the cosmic living organism on the complex-plane section. **[Verification layer]** The five constants (0, 1, \(i\), \(\pi\), \(e\)) correspond to the five core topological nodes of the self-inspection closed loop of the \(T^{64}\) torus, with no superfluous parameters. **Core conclusions**:- **Weak conclusion (can stand independently)**: Euler’s identity may serve as a structural analogy of the cosmic self-inspection closed loop;- **Strong conclusion (depends on the manifestation layer)**: Euler’s identity is the simplest manifestation of cosmic self-inspection on the complex plane. The paper employs a three-tier status system (Frozen / Established within the framework / Analogy), subjects itself to an eight-point meta-cognitive stress test, and supplies explicit falsifiability and verifiability conditions. The correspondence \(35=\) half-turn \(+1\) freezing step and the derivation \(G_{\mathrm{twist}}=\varphi^{-1}\) from the compression iteration \(x_{n+1}=1/(1+x_n)\) are established within the \(T^{64}\) framework. --- **【显影层】** 宇宙是生命体。 **【验证层】** 欧拉恒等式 \(e^{i\pi}+1=0\) 是宇宙生命体拓扑自检在复平面截面上的最简洁显影。 **【验证层】** 五个常数(0、1、\(i\)、\(\pi\)、\(e\))对应 \(T^{64}\) 环面自检闭环的五个核心拓扑节点,没有任何多余参数。 **核心结论**:- **弱结论(可独立成立)**:欧拉恒等式可作为宇宙自检闭环的结构类比;- **强结论(依赖显影层)**:欧拉恒等式是宇宙自检在复平面上的最简显影。 本文采用三档状态标记(已冻结 / 框架内已证 / 类比),先进行八组元认知压力测试,并给出明确的可证伪与可验证条件。\(35=\) 半程 \(+1\) 冻结步的对应,以及由压缩迭代 \(x_{n+1}=1/(1+x_n)\) 导出 \(G_{\mathrm{twist}}=\varphi^{-1}\),均在 \(T^{64}\) 框架内成立。