自然常数 e 的五维跨域映射 | Five faces of one constant, five domains, one skeleton
e 由一个唯一的性质定义：
d(eˣ)/dx = eˣ
唯一一个变化率等于自身的函数。种群增长、放射性衰变、RC 电路、复利、扩散、信息熵——它们都内嵌 e，因为它们共享同一个数学骨架：当前值 → 决定 → 变化率 → 决定 → 下一个值。
e 是这个自缩放结构的指纹。一旦你看到它，它在哪都能看到。
| 维度 | 物理系统 | 公式（骨架） | e 的语义 | 偏离信号 |
|---|---|---|---|---|
| 增长标准 | 复利、衰变 | A(t) = A₀·e^{kt} | 偏离指数 = 检测到约束 | 饱和、阻力 |
| 不确定形状 | 误差、扩散、CLT | e^{−x²} 核 | 随机涨落的普适形状 | 偏度/峰度 ≠ 0 |
| 系统本征函数 | 电路、弹簧、LTI | e^{st} | 线性系统的自然基底 | 非线性残差 |
| 不确定单位 | 玻尔兹曼、softmax | P∝e^{-E/kT} | 信息的自然单位（nat） | 非玻尔兹曼 |
| 复相位（新） | 傅里叶、量子、波动 | e^{iθ}=cosθ+isinθ | 桥接 e 和 π | 非平稳、非简谐 |
e-as-growth-constant/
├── README.md 本文件（双语）
├── SKILL.md OpenClaw skill 定义
├── dimensions/ 五维深度文档
│ ├── 01_continuous_compounding.md
│ ├── 02_normal_distribution.md
│ ├── 03_linear_systems.md
│ ├── 04_information_entropy.md
│ └── 05_complex_phase.md
├── scripts/ 五个可运行 demo
│ ├── compound_growth_demo.py 指数增长 → e / 饱和
│ ├── normal_dist_demo.py CLT / e^{-x²}
│ ├── linear_systems_demo.py RC / 阻尼 / 本征函数
│ ├── entropy_demo.py MaxEnt → 玻尔兹曼
│ └── fourier_bridge_demo.py e^{iθ} / FFT / π 桥接
├── experiments/ 扩展实验
│ ├── iss_analysis.py ISS 轨道衰减分析
│ └── u1_iss_coupling.py U(1) 太阳相位耦合
├── knowledge_graph.md e 的深层数学结构图
├── practical_applications.md 现实世界中的 e
├── test_e_project.py 20 个 pytest 测试用例
├── figures/ 生成的可视化图表
│ └── animation/ 傅里叶桥接动图帧（60帧）
└── LICENSE (MIT)
每个维度都起源于 e 的一个性质：自缩放。不是近似。不是极限。就是自己，在每一点上。
其它的一切都是推论。
e is defined by one property that no other number shares:
d(eˣ)/dx = eˣ
The only function whose rate of change equals itself. Population growth, radioactive decay, RC circuits, compound interest, diffusion, information entropy — they all embed e because they share one mathematical skeleton: current value → determines → rate of change → determines → next value.
e is the fingerprint of this self-scaling structure. Once you see it, you see it everywhere.
| Dimension | Physical System | Formula | e's Semantic Role | Deviation Signal |
|---|---|---|---|---|
| Growth Standard | Compounding, decay | A(t) = A₀·e^{kt} | Deviation from exponential = constraint | Saturation |
| Uncertainty Shape | Error, diffusion, CLT | e^{−x²} kernel | Universal shape of random fluctuation | Skew/kurtosis ≠ 0 |
| System Eigenfunction | Circuits, springs, LTI | e^{st} | Natural basis for linear systems | Nonlinear residue |
| Uncertainty Unit | Boltzmann, softmax | P∝e^{-E/kT} | Natural unit of info (nat) | Non-Boltzmann |
| Complex Phase (NEW) | Fourier, QM, waves | e^{iθ}=cosθ+isinθ | Bridges e and π | Non-stationary |
e-as-growth-constant/
├── README.md Bilingual (this file)
├── SKILL.md OpenClaw skill definition
├── dimensions/ Five deep-dive docs
├── scripts/ Five runnable demos
├── experiments/ Extended experiments
├── knowledge_graph.md e's structural map
├── practical_applications.md e in the real world
├── test_e_project.py 20 pytest cases
├── figures/ Generated visualizations
│ └── animation/ Fourier bridge frames (60)
└── LICENSE (MIT)
Every dimension starts from e's defining property: self-scaling. Not approximately. Not in the limit. Exactly itself, at every point.
Everything else is a consequence.