|
| 1 | +--- |
| 2 | +sidebar_position: 15 |
| 3 | +sidebar_label: Sacred Constants |
| 4 | +--- |
| 5 | + |
| 6 | +# Sacred Constants Reference |
| 7 | + |
| 8 | +Complete catalog of mathematical, physical, and exotic constants available in TRI CLI. All constants are defined in `src/tri/tri_math.zig` and accessible via `tri constants` and `tri math` subcommands. |
| 9 | + |
| 10 | +## Quick Access |
| 11 | + |
| 12 | +```bash |
| 13 | +tri constants # Show core sacred constants (4 sections) |
| 14 | +tri math exotic # Exotic mathematical constants |
| 15 | +tri math physical # Physics constants |
| 16 | +tri math chaos # Chaos theory constants |
| 17 | +tri math all # Display ALL constants |
| 18 | +``` |
| 19 | + |
| 20 | +## `tri constants` Output (Cycle 82) |
| 21 | + |
| 22 | +The `tri constants` command displays 4 sections of constants: |
| 23 | + |
| 24 | +### Section 1: Golden Ratio |
| 25 | + |
| 26 | +| Constant | Symbol | Value | Formula | |
| 27 | +|----------|--------|-------|---------| |
| 28 | +| Golden Ratio | $\phi$ | 1.6180339887498948482 | $(1 + \sqrt{5}) / 2$ | |
| 29 | +| Phi Squared | $\phi^2$ | 2.6180339887498948482 | $\phi^2 = \phi + 1$ | |
| 30 | +| Inverse Phi Squared | $1/\phi^2$ | 0.3819660112501051518 | $1/\phi^2 = \phi - 1$ | |
| 31 | +| Trinity Sum | $\phi^2 + 1/\phi^2$ | 3.0 | TRINITY IDENTITY — exact equality | |
| 32 | + |
| 33 | +**Trinity Identity:** |
| 34 | + |
| 35 | +$$\phi^2 + \frac{1}{\phi^2} = 3 = \text{TRINITY}$$ |
| 36 | + |
| 37 | +### Section 2: Transcendental |
| 38 | + |
| 39 | +| Constant | Symbol | Value | Formula | |
| 40 | +|----------|--------|-------|---------| |
| 41 | +| Pi | $\pi$ | 3.14159265358979323846 | $C / d$ | |
| 42 | +| Euler's Number | $e$ | 2.71828182845904523536 | $\lim_{n\to\infty}(1 + 1/n)^n$ | |
| 43 | +| Transcendental Product | $\pi \times \phi \times e$ | 13.816890703380645 | $\approx$ TRYTE\_MAX (13) | |
| 44 | + |
| 45 | +### Section 3: Genetic Algorithm |
| 46 | + |
| 47 | +| Constant | Symbol | Value | Formula | Description | |
| 48 | +|----------|--------|-------|---------|-------------| |
| 49 | +| Mu | $\mu$ | 0.0382 | $1/\phi^2/10$ | Mutation rate | |
| 50 | +| Chi | $\chi$ | 0.0618 | $1/\phi/10$ | Crossover rate | |
| 51 | +| Sigma | $\sigma$ | 1.618 | $\phi$ | Selection pressure | |
| 52 | +| Epsilon | $\varepsilon$ | 0.333 | $1/3$ | Elitism rate | |
| 53 | + |
| 54 | +### Section 4: Quantum |
| 55 | + |
| 56 | +| Constant | Symbol | Value | Formula | Description | |
| 57 | +|----------|--------|-------|---------|-------------| |
| 58 | +| CHSH Inequality | CHSH | 2.8284271247461903 | $2\sqrt{2}$ | Bell inequality violation | |
| 59 | +| Fine Structure Inverse | $\alpha^{-1}$ | 137.036 | $4\pi^3 + \pi^2 + \pi$ | Fine structure constant | |
| 60 | +| Berry Phase | $\beta$ | 2.112 | $\pi(1 - 1/\phi)$ | Quantum-inspired phase | |
| 61 | +| SU3 Constant | SU3 | 0.927 | $3/(2\phi)$ | Energy harvesting constant | |
| 62 | + |
| 63 | +## Exotic Mathematical Constants (Cycle 83) |
| 64 | + |
| 65 | +```bash |
| 66 | +tri math exotic |
| 67 | +``` |
| 68 | + |
| 69 | +| Constant | Value | Description | |
| 70 | +|----------|-------|-------------| |
| 71 | +| Apery's constant $\zeta(3)$ | 1.2020569031595942 | Sum $\sum_{n=1}^{\infty} 1/n^3$, proved irrational by Apery (1978) | |
| 72 | +| Catalan's constant $G$ | 0.9159655941772190 | $\sum_{n=0}^{\infty} (-1)^n / (2n+1)^2$ | |
| 73 | +| Feigenbaum delta $\delta$ | 4.6692016091029906 | Period-doubling bifurcation ratio (chaos theory) | |
| 74 | +| Feigenbaum alpha $\alpha$ | 2.5029078750958928 | Scaling factor in bifurcation diagram | |
| 75 | +| Khinchin's constant $K$ | 2.6854520010653064 | Geometric mean of continued fraction coefficients | |
| 76 | +| Glaisher-Kinkelin $A$ | 1.2824271291006226 | Related to Barnes G-function | |
| 77 | +| Omega constant $\Omega$ | 0.5671432904097838 | $W(1)$ where $W$ is Lambert W function ($xe^x = 1$) | |
| 78 | +| Plastic number $\rho$ | 1.3247179572447460 | Real root of $x^3 = x + 1$ (cubic golden ratio) | |
| 79 | +| Landau-Ramanujan | 0.7642236535892206 | Density of sums of two squares | |
| 80 | +| Conway's constant $\lambda$ | 1.3035772690342963 | Growth rate of look-and-say sequence | |
| 81 | + |
| 82 | +## Physics Constants (Cycle 83) |
| 83 | + |
| 84 | +```bash |
| 85 | +tri math physical |
| 86 | +``` |
| 87 | + |
| 88 | +### Quantum & Electromagnetic |
| 89 | + |
| 90 | +| Constant | Symbol | Value | Units | |
| 91 | +|----------|--------|-------|-------| |
| 92 | +| Fine structure | $\alpha$ | 0.0072973525693 | dimensionless | |
| 93 | +| Fine structure inverse | $1/\alpha$ | 137.036 | dimensionless | |
| 94 | +| CHSH bound | $2\sqrt{2}$ | 2.8284271247461903 | Bell test maximum | |
| 95 | +| Elementary charge | $e$ | $1.602176634 \times 10^{-19}$ | C | |
| 96 | +| Speed of light | $c$ | 299,792,458 | m/s | |
| 97 | + |
| 98 | +### Planck Units |
| 99 | + |
| 100 | +| Constant | Symbol | Value | Units | |
| 101 | +|----------|--------|-------|-------| |
| 102 | +| Planck constant | $h$ | $6.62607015 \times 10^{-34}$ | J·s | |
| 103 | +| Reduced Planck | $\hbar$ | $1.054571817 \times 10^{-34}$ | J·s | |
| 104 | +| Planck length | $l_P$ | $1.616255 \times 10^{-35}$ | m | |
| 105 | +| Planck time | $t_P$ | $5.391247 \times 10^{-44}$ | s | |
| 106 | +| Planck mass | $m_P$ | $2.176434 \times 10^{-8}$ | kg | |
| 107 | + |
| 108 | +### Thermodynamics & Gravity |
| 109 | + |
| 110 | +| Constant | Symbol | Value | Units | |
| 111 | +|----------|--------|-------|-------| |
| 112 | +| Boltzmann | $k_B$ | $1.380649 \times 10^{-23}$ | J/K | |
| 113 | +| Gravitational | $G$ | $6.67430 \times 10^{-11}$ | m$^3$/(kg·s$^2$) | |
| 114 | +| Avogadro's number | $N_A$ | $6.02214076 \times 10^{23}$ | 1/mol | |
| 115 | + |
| 116 | +## Golden Function Constants (Cycle 84) |
| 117 | + |
| 118 | +Based on Pellis 2025: "The Golden Function and its applications to mathematical physics." |
| 119 | + |
| 120 | +```bash |
| 121 | +tri math golden-function |
| 122 | +``` |
| 123 | + |
| 124 | +| Constant | Value | Description | |
| 125 | +|----------|-------|-------------| |
| 126 | +| $\phi^{1/2}$ | 1.2720196495 | $\sqrt{\phi}$ | |
| 127 | +| $G(0.5)$ | 2.0581710272 | $\sqrt{\phi} + 1/\sqrt{\phi}$ | |
| 128 | +| $\phi^{\phi}$ | 2.3903891399 | Golden self-exponentiation | |
| 129 | +| $\phi^{\pi}$ | 4.5310082907 | Transcendental golden power | |
| 130 | + |
| 131 | +The **Golden Function** $G(x) = \phi^x + \phi^{-x}$ is the continuous extension of Lucas numbers: $G(n) = L(n)$ for integer $n$. |
| 132 | + |
| 133 | +## Nuclear Fibonacci Constants (Cycle 84) |
| 134 | + |
| 135 | +```bash |
| 136 | +tri math nuclear |
| 137 | +``` |
| 138 | + |
| 139 | +Nuclear shell magic numbers correlate with Fibonacci/Lucas sequences: |
| 140 | + |
| 141 | +| Magic number | Fibonacci/Lucas connection | |
| 142 | +|-------------|---------------------------| |
| 143 | +| 2 | $L(0) = 2$ | |
| 144 | +| 8 | $F(6) = 8$ | |
| 145 | +| 20 | $\approx F(8) - 1$ | |
| 146 | +| 28 | $= L(7) - 1$ | |
| 147 | +| 50 | $\approx F(10) - 5$ | |
| 148 | +| 82 | stable shell | |
| 149 | +| 126 | stable shell | |
| 150 | + |
| 151 | +- N/Z stability ratio for heavy nuclei: $\phi / \sqrt{2} \approx 1.144$ |
| 152 | +- Binding energy peak: $8.8 \text{ MeV} \approx F(6) + \phi^{-1}$ |
| 153 | + |
| 154 | +## Chaos Theory (Cycle 83) |
| 155 | + |
| 156 | +```bash |
| 157 | +tri math chaos |
| 158 | +``` |
| 159 | + |
| 160 | +The **Feigenbaum constants** govern the universal behavior of chaotic systems: |
| 161 | + |
| 162 | +- $\delta = 4.669...$: Ratio of successive period-doubling intervals |
| 163 | +- $\alpha = 2.502...$: Scaling factor of function values at bifurcation points |
| 164 | + |
| 165 | +These constants appear in **any** system exhibiting period-doubling route to chaos — from fluid dynamics to population models. |
| 166 | + |
| 167 | +## Command Summary |
| 168 | + |
| 169 | +| Command | Category | Constants shown | |
| 170 | +|---------|----------|----------------| |
| 171 | +| `tri constants` | Core | 4 sections: Golden Ratio, Transcendental, Genetic Algorithm, Quantum | |
| 172 | +| `tri math exotic` | Exotic | 10 rare mathematical constants | |
| 173 | +| `tri math physical` | Physics | 12 fundamental physics constants | |
| 174 | +| `tri math chaos` | Chaos | Feigenbaum + logistic map demo | |
| 175 | +| `tri math all` | Complete | All 76+ constants in catalog | |
| 176 | +| `tri math golden-function` | Advanced | Pellis 2025 continuous Lucas | |
| 177 | +| `tri math nuclear` | Advanced | Nuclear shell magic numbers | |
| 178 | +| `tri math fractal` | Advanced | Fractal dimensions (Sierpinski, Koch, Menger) | |
| 179 | + |
| 180 | +## See Also |
| 181 | + |
| 182 | +- [Sacred Math](/cli/math) — Full math command reference (40+ commands) |
| 183 | +- [Mathematical Foundations](/math-foundations/) — Proofs and theory |
| 184 | +- [Concepts: Trinity Identity](/concepts/trinity-identity) — $\phi^2 + 1/\phi^2 = 3$ proof |
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