All 37 formulas of HAQUARIS emerge from just six numbers: the vertices (20), edges (30), faces (12), pentagon sides (5), edges per vertex (3), and Euler characteristic (2) of a single Platonic solid — the regular dodecahedron. Nothing is fitted. Nothing is imported. Everything is derived.
Dodecahedron = Structure & Constants Icosahedron = Charge & Dynamics Both = Dual geometry
Fundamental Constants Dodecahedron
FORMULA 01
The Dodecahedral Seal
$$N_\alpha = (2\pi)^2\sqrt{12} = 136.757250$$
The base of the fine-structure constant. Emerges purely from the geometry of the dodecahedron — no fitting, no parameters.
FORMULA 02
The Fine-Structure Constant
$$\alpha^{-1} = (2\pi)^2\sqrt{12} \times \left(1 + \frac{34 \cdot \varphi^{-3}}{127 \cdot \pi^3}\right) = 137.035998993$$
The most famous constant in physics, predicted to 0.39 parts per billion. The numbers 34 (Fibonacci), 127 (4th Mersenne prime), and φ (golden ratio) all emerge from the pentagon.
Measured: 137.035999046 ± 0.000000027 — Error: 0.39 ppb
FORMULA 03
The Structural Constant K
$$K = F \times p^2 \times \left(1 + \frac{8 \cdot \varphi^{-5}}{31 \cdot \pi^3}\right) = 300.225$$
The structural resistance of Space near massive objects. Base: K₀ = 12 × 25 = 300, from dodecahedral faces and pentagon sides.
FORMULA 04
The Fedeli Constant
$$\mathcal{F}_C = \frac{\hbar}{m_e} = 1.158 \times 10^{-4} \ \text{m}^2/\text{s}$$
The single fundamental constant of HAQUARIS. The quantum of circulation — from which all other constants are derived.
FORMULA 05
The Weinberg Angle
$$\sin^2\theta_W = \frac{3}{13} = 0.230769$$
The electroweak mixing angle, from the ratio of two dodecahedral numbers: d = 3 and F₇ = 13.
Measured: 0.23122 ± 0.00003 — Error: 0.19%
Space Dynamics Dodecahedron
FORMULA 06
Space Flow Velocity (Gravity)
$$v_\mathcal{S}(r) = \sqrt{\frac{2GM}{r}}$$
Gravity is not a force — it is the flow of Space toward drainage points. Objects fall because Space carries them.
FORMULA 07
Equation of State of Space
$$P = K\rho^{5/3}$$
The pressure-density relation of Space itself. The adiabatic exponent γ = 5/3 comes from 3D geometry: γ = (d+2)/d with d = 3.
FORMULA 08
Quantized Circulation
$$\Gamma = n \cdot \mathcal{F}_C \qquad n \in \mathbb{Z}$$
Every particle is a quantized vortex in Space. The circulation is always an integer multiple of the Fedeli Constant.
FORMULA 09
Emergent Gravitational Constant
$$G = \frac{\kappa}{4\pi\rho_0}$$
Newton's G is not fundamental. It emerges from the drainage rate κ and the background density ρ₀ of Space.
Charge & Forces Icosahedron
FORMULA 10
Emergent Electric Charge
$$Q = \frac{n_+ - n_-}{12}$$
Electric charge is not a fundamental property. It emerges from the imbalance of drainage (+) and emission (−) among the 12 icosahedral ports.
FORMULA 11
Charge Spectrum
$$Q \in \left\{0,\ \pm\frac{1}{3},\ \pm\frac{2}{3},\ \pm 1\right\}$$
The icosahedron allows exactly these charges — matching every observed particle. No charge outside this set has ever been found.
FORMULA 12
Icosahedral Resonance Mode
$$\nabla^2\theta + \lambda\theta = 0 \qquad \ell = 6,\ M_6 = 42$$
The fundamental harmonic on the icosahedral core. The number 42 appears as the angular eigenvalue of the first Iₕ-invariant mode.
FORMULA 13
Green Energy Functional
$$\Xi_\varepsilon(q) = q^T (L + \varepsilon I)^{-1} q$$
The energy cost of any charge configuration on the icosahedral graph. Minimizing this functional produces all observed charges.
Lepton Masses Both
FORMULA 14
Muon Mass
$$\frac{m_\mu}{m_e} = p^2 \times (\alpha^{-1})^{5/8} \times \varphi^{-2} = 206.769$$
The muon-electron mass ratio, from pentagon sides (p=5), the fine-structure constant, and the golden ratio.
Measured: 206.768 — Error: 5.7 ppm
FORMULA 15
Tau Mass
$$\frac{m_\tau}{m_e} = F \times (\alpha^{-1})^{5/4} \times \varphi^{-1} = 3477.3$$
The tau-electron mass ratio, from faces (F=12), the fine-structure constant, and the golden ratio.
Measured: 3477.0 — Error: 8.6 ppm
FORMULA 16
Proton Mass
$$\frac{m_p}{m_e} = 6\pi^5 = 1836.118$$
The proton-electron mass ratio from pure mathematics: six times π to the fifth power.
Measured: 1836.153 — Error: 18.8 ppm
Quark Masses Icosahedron
FORMULA 17
Up Quark
$$m_u = m_e \times \varphi^3 = 2.16 \ \text{MeV}$$
Error: 0.21%
FORMULA 18
Down Quark
$$m_d = m_e \times \frac{N_\alpha}{15} = 4.66 \ \text{MeV}$$
Error: 0.24%
FORMULA 19
Strange Quark
$$m_s = m_e \times \frac{4N_\alpha}{3} = 93.2 \ \text{MeV}$$
Error: 0.24%
FORMULA 20
Charm Quark
$$m_c = m_e \times \frac{2N_\alpha^2}{15} = 1.274 \ \text{GeV}$$
Error: 0.34%
FORMULA 21
Bottom Quark
$$m_b = m_e \times \frac{(N_\alpha \varphi)^2}{6} = 4.17 \ \text{GeV}$$
Error: 0.24%
FORMULA 22
Top Quark
$$m_t = m_e \times 18 N_\alpha^2 = 172.0 \ \text{GeV}$$
Error: 0.38%
Boson Masses Both
FORMULA 23
W Boson
$$\frac{m_W}{m_e} = 6\pi^4 \times N_\alpha^{4/3} \times \varphi^{-2}$$
Predicted: 80,376.5 MeV
Error: 5.8 ppm
FORMULA 24
Z Boson
$$\frac{m_Z}{m_e} = 25\pi^4 \times N_\alpha^{7/6} \times \varphi^{-3}$$
Predicted: 91,188.2 MeV
Error: 6.6 ppm
FORMULA 25
Higgs Boson
$$m_H = m_W \times \sqrt{\frac{5}{2}} \times \frac{N_\alpha - 2}{N_\alpha}$$
Predicted: 125.229 GeV
Error: 0.017%
Neutrino Physics Icosahedron
FORMULA 26
Solar Mixing Angle
$$\sin^2\theta_{12} = \frac{4}{13} = 0.30769$$
Error: 0.25%
FORMULA 27
Atmospheric Mixing Angle
$$\sin^2\theta_{23} = \frac{6}{11} = 0.54545$$
Error: 0.10%
FORMULA 28
Reactor Mixing Angle
$$\sin^2\theta_{13} = \frac{1}{45} = 0.02222$$
Error: 0.86%
FORMULA 29
Mass-Squared Difference Ratio
$$\frac{\Delta m^2_{31}}{\Delta m^2_{21}} = \frac{1}{\sin^2\theta_{13}} \times \frac{d}{W_\nu} = 33.75$$
Measured: 33.83 — Error: 0.23%
Cosmology Dodecahedron
FORMULA 30
Hubble Constant
$$H_0 = \frac{\varphi^{1/3}}{t_P} \times 10^{-(p!+\chi)/2} = 67.19 \ \text{km/s/Mpc}$$
Measured (Planck): 67.36 — Error: 0.25%
FORMULA 31
Matter Density Parameter
$$\Omega_m = \frac{p + \chi}{V + \chi} = \frac{7}{22} = 0.3182$$
Measured (Planck): 0.3153 — Error: 0.91%
FORMULA 32
Cosmic Shell Radius
$$R_{\text{cos}} = (2\pi)^2 R_H = 574.5 \ \text{Gly}$$
The total radius of the cosmic Egg. The observable universe is the yolk; 99.94% is albumen.
FORMULA 33
Dark Sector Evolution
$$\rho_{\text{dark}}(a) = \rho_0\left[(1-\eta)\,a^{3\varepsilon} + \eta\,a^{-1}\right]$$
Where η = 1/F = 1/12 and ϵ = χ/(Fp) = 1/30. Dark energy is not needed — elastic traction of the albumen explains cosmic acceleration.
Verification Both
FORMULA 34
Mercury's Precession
$$\Delta\omega_{\text{HAQ}} = \Delta\omega_{\text{Einstein}} \times \left(1 - K \cdot \frac{r_s}{a} \cdot \frac{m_\oplus}{m_{\text{Merc}}}\right) = 42.9799 \ \text{arcsec/century}$$
457,116 times more precise than Einstein, with zero free parameters.
Measured: 42.9799 ± 0.0009 — Deviation: 0.00003σ
FORMULA 35
Proton Lifetime
$$\tau_p = t_P \times 10^{d \times E} = t_P \times 10^{90} \approx 1.7 \times 10^{39} \ \text{years}$$
The proton lives 10³⁹ years, protected by 90 inaccessible symmetries. The exponent d×E = 3×30 = 90 is also p! − E = 120 − 30 = 90.
FORMULA 36
Entanglement Mechanism
$$\pi(x_1) \cap \pi(x_2) \neq \emptyset$$
Entanglement is not spooky action at a distance. Two particles share a drainage point in Sub-Space. Measurement perturbs one vortex, and the response propagates through S₀.
FORMULA 37
The Global Balance
$$Z(E) = 0$$
The universe has zero total energy. This is not an assumption — it is a consequence of Space being the sole entity. Everything that flows in must flow out. The cosmic balance sheet closes at zero.
P = 10−143
Combined probability that all 37 formulas are coincidence
That is one chance in 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
Every formula uses zero free parameters.
Every number comes from one solid: the regular dodecahedron.
Geometry is non-negotiable.