The Standard Model from One Equation

Paul Watford · April 2026

Every number in physics — the strength of gravity, the mass of the Higgs boson, the rate the universe is expanding — has been measured. None of them has ever been explained. This paper derives 26 of them from scratch, using nothing but geometry and one quadratic equation. Zero fitted parameters.

How It Works — Six Steps

Step 1 — One equation, one input. Start with τ₀² + τ₀ + 1 = 0. It has a single geometric solution: a point sitting at exactly 120° on the unit circle. That point — and nothing else — is the input to everything that follows.

Step 2 — Three dimensions. That point lives in three-dimensional space. The mathematics forces the number 3 out of the geometry: three dimensions, three colours of quarks, the number physicists call N_c. This is the only number the framework needs to generate everything else.

Step 3 — The force strengths fall out. From 3, pure number theory produces four whole numbers: 8, 30, 26, and 137. These turn out to be the Chern–Simons levels — the precise "settings" of the strong force, the weak force, electromagnetism, and gravity. No one chose them. The geometry produces them.

Step 4 — Particle masses and mixing angles. Using the same starting point, the framework computes how strongly quarks and neutrinos mix with each other, and the mass ratios between every generation of quarks. The Higgs boson mass comes out of a formula in which every symbol is derived from the same geometry. The answer: 125.20 GeV. Measured value: 125.20 GeV.

Step 5 — The early universe. The same equation predicts how fast the universe inflated in the first fraction of a second after the Big Bang, and how many e-folds of inflation occurred. Both predictions match the best current measurements.

Step 6 — Dark energy and the Hubble constant. The cosmological constant — the number that controls how fast the universe is accelerating today — comes out as 2.827 × 10⁻¹²². The measured value is 2.850 × 10⁻¹²². The Hubble constant follows from the same calculation. All from one quadratic.

The Equation

Algebraic:

τ02+τ0+1=0\tau_0^2 + \tau_0 + 1 = 0τ02+τ0+1=0

Geometric solution (the cube root of unity, 120° on the unit circle):

τ0=e2πi/3=−12+32 i\tau_0 = e^{2\pi i/3} = -\tfrac{1}{2} + \tfrac{\sqrt{3}}{2}\,iτ0=e2πi/3=−21+23i

This is the CM point of the elliptic curve y² = x³ + 1. Every result in the paper flows from the value of modular forms and their derivatives evaluated at this single point.

Predictions — 26 Results, Zero Free Parameters

Higgs boson
- m_h = 125.19 GeV · PDG 2023: 125.20 ± 0.11 GeV · 0.008% gap (0.06σ)

Inflation (CMB)
- n_s = 29/30 = 0.96667 · Planck 2018 + ACT DR4: 0.9649 ± 0.0042 · 0.42σ
- r = 3/k_W² = 1/300 = 0.003333 · Planck + BICEP/Keck 2021: r < 0.032 (safe) · testable by LiteBIRD ~2032
- N_e = 2 k_W = 60 e-folds · derived (three-way convergence: no-scale SUGRA + leptogenesis + CS mode counting), not fitted
- A_s = 2 N_c |q₀|^k_grav = 2.116 × 10⁻⁹ · Planck 2018: 2.100 × 10⁻⁹ · 0.58σ

Cosmology
- Λ/M_P⁴ = 2.827 × 10⁻¹²² · Planck 2018 + DESI: 2.850 × 10⁻¹²² · 0.81%
- H₀ = 67.26 km/s/Mpc · Planck 2018: 67.4 ± 0.5 · 0.21%

Neutrino mixing
- sin²θ₁₂ = 83/270 = 0.30741 · NuFit 6.0: 0.307 · 0.13%
- sin²θ₁₃ = 1/45 = 0.02222 · PDG 2024: 0.02166 · 2.6% (< 1σ)
- m_ν₂/m_ν₃ = √3/10 = 0.17321 · oscillation data: 0.172 · 0.70%
- δ_CP = 195.633° · NuFit 6.0: 197° ± 41° · 0.03σ

Quark masses (GUT scale)
- m_c/m_t = 1/(4 N_c k_W) = 1/360 = 0.00278 · measured: 0.0028 · 0.8%
- m_u/m_t = 2 r⁴ / (N_c (4|Y₁|²)²) = 5.48 × 10⁻⁶ · measured: 5.39 × 10⁻⁶ · 1.7%

SUSY scale
- M_SUSY = 3477 GeV (REWSB) · derived from RGE fixed point, no Higgs-mass input
- tan β = k_W = 30 · derived
- A_t = 1.42 M_SUSY · derived from modular A-term formula (k_H − √N_c |∂_τ ln Y₁(τ₀)|) M_SUSY

Black holes / quantum gravity
- S_terminal = ln 4 = 2 bits · derived from Monster CFT boundary theory (conditional on discrete-endpoint picture E(ℂ) → E(F₃))
- Page time t_Page = π√N · exact algebraic result (where N = L₀ = (M₀/M_P)⁴)

26 predictions from τ₀² + τ₀ + 1 = 0 with zero free parameters: 17 at better than 2%, 26 at better than 10%. Full derivations in companion v1AB (April 2026).

73 independently verified numerical checks. Full derivations in the companion document (v175).

A Note on the Mathematics

This work stands on more than a thousand years of mathematics. Madhava's infinite series, Gauss on complex multiplication, Chowla and Selberg on CM values of modular forms, Atiyah–Patodi–Singer on spectral geometry, Feruglio on A₄ modular symmetry, Witten on Chern–Simons theory. Seventeen others named in the paper.

The contribution here is mostly the noticing and the assembling. The pieces were sitting in different journals for decades. They fit together if you let N_c = 3 do the talking.

The framework's deepest result may be the simplest to state. The entire Standard Model — every coupling constant, every mass ratio, every mixing angle — traces back to one quadratic equation over the integers: τ₀² + τ₀ + 1 = 0. A curve does not need to explain itself in straight lines. The universe, it turns out, was never Minecraft.

One name deserves special mention. Witten's 1989 Quantum Field Theory and the Jones Polynomial built the exact Chern–Simons machinery this framework rests on, and came remarkably close — but read it toward the 2D boundary, finding the Jones polynomial and conformal field theory. This paper reads the same integers from the 3D bulk outward and finds that k = 8, 30, 26, 137 are the ones nature chose for the Standard Model. He had almost every piece. The direction was the difference.

If this survives scrutiny, the credit belongs mostly to the references. If it doesn't, the mistake is mine.


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