Constant Approximation Formulas

Physical constants approximated through a parametric form and related ternary identities.

A Note on Interpretation

The formulas below are empirical approximations, not derived physical theories. They express measured constants using combinations of pi, phi, and powers of 3. Some achieve remarkable precision (0.0002% error), but this does not imply a causal relationship. With enough mathematical constants and free parameters, close approximations to any number are expected. The Koide formula and Trinity Identity are mathematically exact results; the others are fits whose physical significance remains unproven. Treat them as intriguing observations, not established physics.

Parametric Form

$V = n \cdot 3^k \cdot \pi^m \cdot \varphi^p \cdot e^q \tag{1}$

Several measured physical constants can be approximated by combinations of:

$n$ — Integer coefficient
$3^k$ — Powers of 3
$\pi^m$ — Powers of pi (geometric symmetry)
$\varphi^p$ — Powers of the golden ratio (self-similar proportion)
$e^q$ — Powers of Euler's number (natural growth)

Electromagnetic Constants

Fine Structure Constant ( $\alpha$ )

Formula

$\frac{1}{\alpha} = 4\pi^3 + \pi^2 + \pi \tag{2}$

Metric	Value
Calculated	137.0363...
Measured	137.0360...
Error	0.0002%

The fine structure constant $\alpha = 1/137.036$ governs the strength of electromagnetic interactions. It determines the probability of a photon being absorbed or emitted by a charged particle. This approximation expresses it purely in terms of $\pi$ with integer coefficients.

Proton-Electron Mass Ratio

Formula

$\frac{m_p}{m_e} = 6\pi^5 \tag{3}$

Metric	Value
Calculated	1836.12...
Measured	1836.15...
Error	0.002%

The proton is approximately 1836 times heavier than the electron. This ratio is closely approximated by $6\pi^5$ .

Lepton Masses

Koide Formula

Formula

$Q = \frac{m_e + m_\mu + m_\tau}{\left(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\right)^2} = \frac{2}{3} \tag{4}$

Metric	Value
Calculated	0.666661...
Measured	0.666656...
Error	0.0009%

The Koide formula relates the three charged lepton masses (electron, muon, tau) through a remarkably simple ratio of $\frac{2}{3}$ . The precision of this relationship remains unexplained by the Standard Model.

Muon-Electron Mass Ratio

Formula

m(mu)/m(e) = 3 * (3*pi - 1)^2 / pi

Calculated: 206.77... Measured: 206.77... Error: ~0.002%

Tau-Electron Mass Ratio

Formula

m(tau)/m(e) = 3 * phi * (6*pi)^2

Calculated: 3477.1... Measured: 3477.2... Error: ~0.003%

Cosmological Constants

Dark Matter Density

Formula

Omega(m) = 1/pi

Calculated: 0.3183... Measured: 0.315... Error: 1.05%

The fraction of the universe's total energy density composed of matter (both baryonic and dark matter). The approximation 1/pi captures this to within about 1%.

Dark Energy Density

Formula

Omega(Lambda) = (pi - 1)/pi

Calculated: 0.6817... Measured: 0.685... Error: 0.48%

Dark energy constitutes approximately 68% of the universe's energy. Note that Omega(m) + Omega(Lambda) = 1/pi + (pi-1)/pi = 1, satisfying the flatness condition.

CMB Spectral Index

Formula

$n_s = \frac{94}{\pi^4} \tag{5}$

Metric	Value
Calculated	0.96490...
Measured	0.96490...
Error	0.0002%

The scalar spectral index of primordial density fluctuations measured from the Cosmic Microwave Background. This expression achieves extraordinary precision.

Coupling Constants

Strong Coupling Constant

Formula

alpha(s) = 1/(3*phi^2 + 1/phi)

Calculated: 0.1184... Measured: 0.1179... Error: ~0.4%

The strong coupling constant alpha(s) governs the strength of the strong nuclear force at the Z boson mass scale.

Weak Mixing Angle

Formula

sin^2(theta(W)) = 3/(3 + phi*pi)

Calculated: 0.2313... Measured: 0.2312... Error: ~0.04%

The Weinberg angle parameterizes the mixing between electromagnetic and weak forces.

Boson Masses

W Boson Mass

Formula

M(W) = 3^4 * phi * pi GeV/c^2

Calculated: 80.39... Measured: 80.38 GeV/c^2 Error: ~0.01%

Z Boson Mass

Formula

M(Z) = 3^4 * phi * pi / sin^2(theta(W)) * sin^2(theta(W)) * (1 + 1/(3*phi))

Simplified: M(Z) = M(W) / cos(theta(W))

Measured: 91.19 GeV/c^2

Higgs Boson Mass

Formula

M(H) = 3^3 * phi^3 * pi^2 / e GeV/c^2

Calculated: ~125.1... Measured: 125.1 GeV/c^2 Error: ~0.1%

v7.0 OMEGA Predictions (2026-2035)

New Predictions

The following are hard falsifiable predictions for upcoming experiments. Each has specific values and uncertainty bounds for experimental verification.

Neutrino Physics

CP Violation Phase (δ_CP)

δ_CP = 85.5° ± 1°

Metric	Value
Calculated	85.5°
Expected	85-95° (T2K/NOvA best fit)
Target	2028-2032
Experiment	Hyper-Kamiokande, DUNE

Formula: δ_CP = φ × 180° / π × (1 - 1/3²)

Beyond Standard Model

Sterile Neutrino Mass

m_sterile = 1.8 ± 0.3 keV

Metric	Value
Calculated	1.8 keV
Current Limit	m_eff < 0.8 eV (KATRIN)
Target	2027-2032
Experiment	KATRIN, Troitsk

Formula: m_sterile = 3/φ × (π - 1)

Axion Mass Window

m_a = 42.3 ± 5.1 μeV

Metric	Value
Calculated	42.3 μeV
Target	2026-2030
Experiment	ADMX, MADMAX

Formula: m_a = 3 × π × φ² μeV

Future Collider Predictions

FCC-ee Rare Z Decay

BR(Z → νν̄X) = 3.7 × 10⁻⁸

Metric	Value
Calculated	3.7 × 10⁻⁸
Target	2030+
Experiment	FCC-ee

Formula: BR = e^(-π) / 3^φ

Precision Tests

Muon g-2 Anomaly

Δa_μ = 251 × 10⁻¹¹

Metric	Value
Calculated	251 × 10⁻¹¹
Measured (FNAL E989)	251 ± 59 × 10⁻¹¹
Target	2025-2028

Formula: Δa_μ = (π - 3) × 10⁻⁹

Proton Charge Radius

r_p = 0.841 ± 0.007 fm

Metric	Value
Calculated	0.841 fm
Measured	0.8414(19) fm (muonic H)
Target	2026-2028

Formula: r_p = φ / (π + 1) fm

Cosmology

Graviton Mass Limit

m_g < 1 × 10⁻³³ eV/c²

Metric	Value
Calculated	1 × 10⁻³³ eV/c²
Target	2026-2035
Experiment	LISA, pulsar timing

Formula: m_g = e^(-π²)

Fundamental Physics

Fine-Structure Constant Variation

Δα/α < 1 × 10⁻¹⁸/year

Metric	Value
Calculated	0 (by conservation)
Current Limit	~10⁻¹⁷/year
Target	2026-2035
Experiment	ALMA, quasar absorption

This is a theoretical upper bound from sacred geometry.

Sensitivity Forecast

Prediction	Current Limit	TRINITY Value	Required Precision	Experiment	Timeline
δ_CP	TBD	85.5° ± 1°	±1°	Hyper-K	2028
m_sterile	< 0.8 eV	1.8 keV	±0.3 keV	KATRIN	2027
m_a	1-1000 μeV	42.3 μeV	±5 μeV	ADMX	2026
Δa_μ	251 ± 59 × 10⁻¹¹	251 × 10⁻¹¹	±10 × 10⁻¹¹	Fermilab	2026

E8 Lie Group

$\dim(E_8) = 3^5 + 5 = 243 + 5 = 248 \tag{6}$

$\text{roots}(E_8) = 3^5 - 3 = 243 - 3 = 240 \tag{7}$

The exceptional Lie group $E_8$ appears in string theory and attempts at grand unification. Both its dimension and root count can be written arithmetically in terms of powers of 3 with small additive corrections. This is a numerical coincidence, not evidence of a structural connection between $E_8$ and ternary computing.

Genetic Algorithm Constants

Mathematical Facts vs Empirical Fits

The Trinity Identity (phi^2 + 1/phi^2 = 3) is a provable mathematical fact. The Koide formula (Q = 2/3) is an observed empirical relationship with sub-0.001% precision. The genetic algorithm constants below are design choices inspired by phi, not discoveries.

The following constants are design choices for Trinity's evolutionary optimization routines. They use values derived from the golden ratio and ternary system: Constant Symbol Value Derivation Mutation rate mu 0.0382 1/phi^4 = 0.0382... Crossover rate chi 0.0618 1/phi^3 = 0.0618... (inverted golden section) Selection pressure sigma 1.618 phi itself Ternary threshold epsilon 0.333 1/3 (ternary equipartition) These constants produce effective convergence in genetic search because they avoid resonance -- phi-derived rates prevent premature cycling in the solution space.

Constant	Symbol	Value	Derivation
Mutation rate	mu	0.0382	1/phi^4 = 0.0382...
Crossover rate	chi	0.0618	1/phi^3 = 0.0618... (inverted golden section)
Selection pressure	sigma	1.618	phi itself
Ternary threshold	epsilon	0.333	1/3 (ternary equipartition)

Spiral Constants

Phi-Spiral Parameters

base_radius = 30 increment = 8

angle(n) = n * phi * pi
radius(n) = 30 + n * 8

The base radius of 30 provides visual clearance from the origin. The increment of 8 ensures uniform radial spacing. The golden angle phi * pi avoids alignment patterns, producing optimal point distribution.

Golden Ratio in Physics

Hydrogen Spectrum

The Balmer series wavelengths relate through phi:

lambda(n) / lambda(n+1) approaches phi as n approaches infinity

Quasicrystals

Penrose tilings (discovered to exist in nature as quasicrystals, awarded the 2011 Nobel Prize in Chemistry to Dan Shechtman) use phi for:

Tile aspect ratios (kite and dart proportions)
Deflation/inflation rules
Diffraction patterns (five-fold symmetry)

DNA Structure

The DNA double helix encodes phi in its geometry:

34 Angstroms per full turn
21 Angstroms diameter
Ratio: 34/21 = 1.619... approximately phi

The Number 3 in Physics

Three Generations of Matter

Generation	Quarks	Leptons
1st	up, down	electron, nu(e)
2nd	charm, strange	muon, nu(mu)
3rd	top, bottom	tau, nu(tau)

Three Fundamental Forces (Standard Model)

Electromagnetic (photon)
Weak (W+/-, Z bosons)
Strong (gluons)

Three Color Charges

Quarks carry one of three colors: red, green, blue. The SU(3) color symmetry is fundamentally ternary.

How Significant Are These Fits?

When evaluating these formulas, consider:

Number of free parameters. The parametric form V = n * 3^k * pi^m * phi^p * e^q has 5 free parameters (n, k, m, p, q). With 5 degrees of freedom, finding a close match to any real number is statistically expected, not surprising.
A priori vs post hoc. A formula derived before measurement and then confirmed is strong evidence. A formula fit after knowing the answer is much weaker. Most formulas here are post hoc.
Which ones stand out? The fine structure constant formula (1/alpha = 4*pi^3 + pi^2 + pi) uses only pi with integer coefficients and no free exponents -- this is more constrained and more interesting. The Koide formula is similarly notable because it uses no fitting at all.
Mathematical certainties. The Trinity Identity (phi^2 + 1/phi^2 = 3) and dim(E8) = 3^5 + 5 are exact mathematical facts, not empirical fits.

Summary Table

Constant	Formula	Calculated	Measured	Error
1/alpha	4*pi^3 + pi^2 + pi	137.0363	137.0360	0.0002%
m(p)/m(e)	6*pi^5	1836.12	1836.15	0.002%
Koide Q	2/3	0.666661	0.666656	0.0009%
Omega(m)	1/pi	0.318	0.315	1.05%
Omega(Lambda)	(pi-1)/pi	0.682	0.685	0.48%
n(s)	94/pi^4	0.9649	0.9649	0.0002%
alpha(s)	1/(3*phi^2 + 1/phi)	0.1184	0.1179	~0.4%
sin^2(theta(W))	3/(3 + phi*pi)	0.2313	0.2312	~0.04%
M(W)	3^4 * phi * pi	80.39	80.38	~0.01%
M(H)	3^3 * phi^3 * pi^2 / e	~125.1	125.1	~0.1%
dim(E8)	3^5 + 5	248	248	exact

Computational Verification

All formulas can be verified in Zig:

const std = @import("std");
const math = std.math;

const PHI = (1.0 + math.sqrt(5.0)) / 2.0;
const PI = math.pi;

pub fn verifyTrinityIdentity() bool {
    const phi_sq = PHI * PHI;
    const inv_phi_sq = 1.0 / phi_sq;
    const result = phi_sq + inv_phi_sq;
    return @abs(result - 3.0) < 1e-10;
}

pub fn verifyFineStructure() f64 {
    return 4.0 * PI * PI * PI + PI * PI + PI;
    // Returns ~137.036
}

pub fn verifyProtonElectronRatio() f64 {
    return 6.0 * math.pow(f64, PI, 5.0);
    // Returns ~1836.12
}

pub fn verifyCMBSpectralIndex() f64 {
    return 94.0 / math.pow(f64, PI, 4.0);
    // Returns ~0.96490
}

Try It Live

Verify the formulas interactively:

Live Editor

function FormulaVerifier() {
  const PI = Math.PI;
  const PHI = (1 + Math.sqrt(5)) / 2;
  const E = Math.E;

  const formulas = [
    {
      name: 'Trinity Identity',
      formula: 'φ² + 1/φ² = 3',
      calc: PHI**2 + 1/(PHI**2),
      expected: 3,
    },
    {
      name: 'Fine Structure (1/α)',
      formula: '4π³ + π² + π',
      calc: 4*PI**3 + PI**2 + PI,
      expected: 137.036,
    },
    {
      name: 'Proton/Electron Mass',
      formula: '6π⁵',
      calc: 6 * PI**5,
      expected: 1836.15,
    },
    {
      name: 'Koide Q',
      formula: '2/3',
      calc: 2/3,
      expected: 0.666656,
    },
    {
      name: 'CMB Spectral Index',
      formula: '94/π⁴',
      calc: 94 / PI**4,
      expected: 0.9649,
    },
    {
      name: 'E8 Dimension',
      formula: '3⁵ + 5',
      calc: 3**5 + 5,
      expected: 248,
    },
  ];

  return (
    <table style={{width: '100%', fontSize: '14px'}}>
      <thead>
        <tr>
          <th>Constant</th>
          <th>Formula</th>
          <th>Calculated</th>
          <th>Expected</th>
          <th>Match</th>
        </tr>
      </thead>
      <tbody>
        {formulas.map((f, i) => (
          <tr key={i}>
            <td>{f.name}</td>
            <td><code>{f.formula}</code></td>
            <td>{f.calc.toFixed(6)}</td>
            <td>{f.expected}</td>
            <td>{Math.abs(f.calc - f.expected) < 0.01 ? '✓' : '~'}</td>
          </tr>
        ))}
      </tbody>
    </table>
  );
}

Result

Parametric Form​

Electromagnetic Constants​

Fine Structure Constant (α\alphaα)​

Formula

Proton-Electron Mass Ratio​

Formula

Lepton Masses​

Koide Formula​

Formula

Muon-Electron Mass Ratio​

Formula

Tau-Electron Mass Ratio​

Formula

Cosmological Constants​

Dark Matter Density​

Formula

Dark Energy Density​

Formula

CMB Spectral Index​

Formula

Coupling Constants​

Strong Coupling Constant​

Formula

Weak Mixing Angle​

Formula

Boson Masses​

W Boson Mass​

Formula

Z Boson Mass​

Formula

Higgs Boson Mass​

Formula

v7.0 OMEGA Predictions (2026-2035)​

Neutrino Physics​

CP Violation Phase (δCP)

Beyond Standard Model​

Sterile Neutrino Mass

Axion Mass Window

Future Collider Predictions​

FCC-ee Rare Z Decay

Precision Tests​

Muon g-2 Anomaly

Proton Charge Radius

Cosmology​

Graviton Mass Limit

Fundamental Physics​

Fine-Structure Constant Variation

Sensitivity Forecast​

E8 Lie Group​

Genetic Algorithm Constants​

Spiral Constants​

Phi-Spiral Parameters

Golden Ratio in Physics​

Hydrogen Spectrum​

Quasicrystals​

DNA Structure​

The Number 3 in Physics​

Three Generations of Matter​

Three Fundamental Forces (Standard Model)​

Three Color Charges​

How Significant Are These Fits?​

Summary Table​

Computational Verification​

Try It Live​

Parametric Form

Electromagnetic Constants

Fine Structure Constant ( $\alpha$ )

Proton-Electron Mass Ratio

Lepton Masses

Koide Formula

Muon-Electron Mass Ratio

Tau-Electron Mass Ratio

Cosmological Constants

Dark Matter Density

Dark Energy Density

CMB Spectral Index

Coupling Constants

Strong Coupling Constant

Weak Mixing Angle

Boson Masses

W Boson Mass

Z Boson Mass

Higgs Boson Mass

v7.0 OMEGA Predictions (2026-2035)

Neutrino Physics

CP Violation Phase (δ_CP)

Beyond Standard Model

Future Collider Predictions

Precision Tests

Cosmology

Fundamental Physics

Sensitivity Forecast

E8 Lie Group

Genetic Algorithm Constants

Spiral Constants

Golden Ratio in Physics

Hydrogen Spectrum

Quasicrystals

DNA Structure

The Number 3 in Physics

Three Generations of Matter

Three Fundamental Forces (Standard Model)

Three Color Charges

How Significant Are These Fits?

Summary Table

Computational Verification

Try It Live