Sacred Constants Reference

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.

Quick Access

tri constants           # Show core sacred constants (4 sections)
tri math exotic         # Exotic mathematical constants
tri math physical       # Physics constants
tri math chaos          # Chaos theory constants
tri math all            # Display ALL constants

tri constants Output (Cycle 82)

The tri constants command displays 4 sections of constants:

Section 1: Golden Ratio

Constant	Symbol	Value	Formula
Golden Ratio	$\phi$	1.6180339887498948482	$(1 + \sqrt{5}) / 2$
Phi Squared	$\phi^2$	2.6180339887498948482	$\phi^2 = \phi + 1$
Inverse Phi Squared	$1/\phi^2$	0.3819660112501051518	$1/\phi^2 = \phi - 1$
Trinity Sum	$\phi^2 + 1/\phi^2$	3.0	TRINITY IDENTITY — exact equality

Trinity Identity:

$\phi^2 + \frac{1}{\phi^2} = 3 = \text{TRINITY}$

Section 2: Transcendental

Constant	Symbol	Value	Formula
Pi	$\pi$	3.14159265358979323846	$C / d$
Euler's Number	$e$	2.71828182845904523536	$\lim_{n\to\infty}(1 + 1/n)^n$
Transcendental Product	$\pi \times \phi \times e$	13.816890703380645	$\approx$ TRYTE_MAX (13)

Section 3: Genetic Algorithm

Constant	Symbol	Value	Formula	Description
Mu	$\mu$	0.0382	$1/\phi^2/10$	Mutation rate
Chi	$\chi$	0.0618	$1/\phi/10$	Crossover rate
Sigma	$\sigma$	1.618	$\phi$	Selection pressure
Epsilon	$\varepsilon$	0.333	$1/3$	Elitism rate

Section 4: Quantum

Constant	Symbol	Value	Formula	Description
CHSH Inequality	CHSH	2.8284271247461903	$2\sqrt{2}$	Bell inequality violation
Fine Structure Inverse	$\alpha^{-1}$	137.036	$4\pi^3 + \pi^2 + \pi$	Fine structure constant
Berry Phase	$\beta$	2.112	$\pi(1 - 1/\phi)$	Quantum-inspired phase
SU3 Constant	SU3	0.927	$3/(2\phi)$	Energy harvesting constant

Exotic Mathematical Constants (Cycle 83)

tri math exotic

Constant	Value	Description
Apery's constant $\zeta(3)$	1.2020569031595942	Sum $\sum_{n=1}^{\infty} 1/n^3$, proved irrational by Apery (1978)
Catalan's constant $G$	0.9159655941772190	$\sum_{n=0}^{\infty} (-1)^n / (2n+1)^2$
Feigenbaum delta $\delta$	4.6692016091029906	Period-doubling bifurcation ratio (chaos theory)
Feigenbaum alpha $\alpha$	2.5029078750958928	Scaling factor in bifurcation diagram
Khinchin's constant $K$	2.6854520010653064	Geometric mean of continued fraction coefficients
Glaisher-Kinkelin $A$	1.2824271291006226	Related to Barnes G-function
Omega constant $\Omega$	0.5671432904097838	$W(1)$ where $W$ is Lambert W function ($xe^x = 1$)
Plastic number $\rho$	1.3247179572447460	Real root of $x^3 = x + 1$ (cubic golden ratio)
Landau-Ramanujan	0.7642236535892206	Density of sums of two squares
Conway's constant $\lambda$	1.3035772690342963	Growth rate of look-and-say sequence

Physics Constants (Cycle 83)

tri math physical

Quantum & Electromagnetic

Constant	Symbol	Value	Units
Fine structure	$\alpha$	0.0072973525693	dimensionless
Fine structure inverse	$1/\alpha$	137.036	dimensionless
CHSH bound	$2\sqrt{2}$	2.8284271247461903	Bell test maximum
Elementary charge	$e$	$1.602176634 \times 10^{-19}$	C
Speed of light	$c$	299,792,458	m/s

Planck Units

Constant	Symbol	Value	Units
Planck constant	$h$	$6.62607015 \times 10^{-34}$	J·s
Reduced Planck	$\hbar$	$1.054571817 \times 10^{-34}$	J·s
Planck length	$l_P$	$1.616255 \times 10^{-35}$	m
Planck time	$t_P$	$5.391247 \times 10^{-44}$	s
Planck mass	$m_P$	$2.176434 \times 10^{-8}$	kg

Thermodynamics & Gravity

Constant	Symbol	Value	Units
Boltzmann	$k_B$	$1.380649 \times 10^{-23}$	J/K
Gravitational	$G$	$6.67430 \times 10^{-11}$	m$^3$/(kg·s$^2$)
Avogadro's number	$N_A$	$6.02214076 \times 10^{23}$	1/mol

Golden Function Constants (Cycle 84)

Based on Pellis 2025: "The Golden Function and its applications to mathematical physics."

tri math golden-function

Constant	Value	Description
$\phi^{1/2}$	1.2720196495	$\sqrt{\phi}$
$G(0.5)$	2.0581710272	$\sqrt{\phi} + 1/\sqrt{\phi}$
$\phi^{\phi}$	2.3903891399	Golden self-exponentiation
$\phi^{\pi}$	4.5310082907	Transcendental golden power

The Golden Function $G(x) = \phi^x + \phi^{-x}$ is the continuous extension of Lucas numbers: $G(n) = L(n)$ for integer $n$.

Nuclear Fibonacci Constants (Cycle 84)

tri math nuclear

Nuclear shell magic numbers correlate with Fibonacci/Lucas sequences:

Magic number	Fibonacci/Lucas connection
2	$L(0) = 2$
8	$F(6) = 8$
20	$\approx F(8) - 1$
28	$= L(7) - 1$
50	$\approx F(10) - 5$
82	stable shell
126	stable shell

• N/Z stability ratio for heavy nuclei: $\phi / \sqrt{2} \approx 1.144$
• Binding energy peak: $8.8 \text{ MeV} \approx F(6) + \phi^{-1}$

Chaos Theory (Cycle 83)

tri math chaos

The Feigenbaum constants govern the universal behavior of chaotic systems:

• $\delta = 4.669...$: Ratio of successive period-doubling intervals
• $\alpha = 2.502...$: Scaling factor of function values at bifurcation points

These constants appear in any system exhibiting period-doubling route to chaos — from fluid dynamics to population models.

Command Summary

Command	Category	Constants shown
tri constants	Core	4 sections: Golden Ratio, Transcendental, Genetic Algorithm, Quantum
tri math exotic	Exotic	10 rare mathematical constants
tri math physical	Physics	12 fundamental physics constants
tri math chaos	Chaos	Feigenbaum + logistic map demo
tri math all	Complete	All 76+ constants in catalog
tri math golden-function	Advanced	Pellis 2025 continuous Lucas
tri math nuclear	Advanced	Nuclear shell magic numbers
tri math fractal	Advanced	Fractal dimensions (Sierpinski, Koch, Menger)

See Also

• Sacred Math — Full math command reference (40+ commands)
• Mathematical Foundations — Proofs and theory
• Concepts: Trinity Identity — $\phi^2 + 1/\phi^2 = 3$ proof