The Standard Model from the CM Point of y² = x³ + 1

The Standard Model from One Equation Chern-Simons Quantization on the Exceptional Jordan Algebra h₃ (O) 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 from first principles. This paper derives 26 of them from scratch, using nothing but geometry and one quadratic equation. Zero fitted parameters. The mathematical structure underneath turns out to be 3D Chern-Simons theory on the exceptional Jordan algebra h₃ (O), with A₄ modular flavor symmetry, anchored at the complex-multiplication point of the elliptic curve y² = x³ + 1. Every Standard Model integer - the force strengths, the generation count, the soft-SUSY mass structure - is a dimension or mode count in this exceptional structure. The contribution of this paper is mostly the noticing: the pieces were published separately over the last fifty years. They fit. HOW IT WORKS - SIX STEPS Step 1 - One equation, one input. Start with tau₀² + tau₀ + 1 = 0. It has a single geometric solution: a point sitting at exactly 120 degrees on the unit circle. This is the CM point of y² = x³ + 1, with Gauss class number h (-3) = 1. That point - and nothing else - is the input to everything that follows. Step 2 - Three dimensions. The mathematics forces (2 Im tau₀) ² = 3. Three dimensions, three colours of quarks, the number physicists call Nc = 3. The stabilizer of tau₀ in PSL (2, Z) is Z/3, which is simultaneously the centre of SU (3), the centre of E₆, and the Z/3 that permutes the three off-diagonal octonion slots of h₃ (O) via SO (8) triality (Dubois-Violette-Todorov 2018). Three is not input three times; it is one three, in three languages. Step 3 - The force strengths are integer channel counts. From Nc = 3, the Chern-Simons Interaction Module Theorem produces four whole numbers: kₛ = 8 = dim O - octonion dimension, coincident with dim adj SU (3) kW = 30 - zeta-regularised Peter-Weyl sum on S³, equivalently P (4) = sum (2j+1) ² for j 125. 97 -> 125. 19 GeV PDG 2023: 125. 20 +/- 0. 11 via the modular A-term Aₜ = sqrt (kH) * MS = 1. 42 MS and a 3-loop QCD coefficient K^ (rho rho) (tau₀) = 1/4 proved by three-way convergence at Nc = 3 (axiom-derived Kahler coefficient, gravitational integer 1/kgrav = 1/|E (F₃) |, and standard QCD TF²). The triple identification is unique to Nc = 3. Step 5 - SUSY breaking as democratic distribution on h₃ (O). The Bridge Theorem (R17, April 2026) derives the soft-mass formula m² (Md, MSUSY) = (d/27) * MSUSY² where d is the dimension of the h₃ (O) sub-module containing the field. Three independent SUGRA Lagrangian sectors - superpotential mu-term, Higgs Kahler metric, trilinear A-term - each yield one constraint on (Nc, kH). Matched against the Dubois-Violette-Todorov sub-module decomposition of hN (O), the three-constraint system has a unique non-trivial integer solution (Nc, kH) = (3, 2): muGM² / M² = 3/27 - dim (diag h₃ (O) ) / 27 mu-term sector, Giudice-Masiero |m²Hu (MSUSY) | / M² = 24/27 - dim (off-diag h₃ (O) ) / 27 focus-point after RGE from Kahler boundary m²Hd (MSUSY) / M² = 18/27 - dim (off-diag O/C) / 27 Kahler metric at MGUT + universality + weak running (|m²Hu| - m²Hd) / M² = 6/27 - dim (off-diag C) / 27 algebraic difference Aₜ² / MS² = 2 = dim (traceless diag h₃ (O) ) A-term sector, this is Step 3 closure Sum: 3/27 + 24/27 = 27/27 = 1 is dimensional completeness of h₃ (O). The Pythagorean identity muGM² + muₜotal² + mZ²/2 = M² becomes a statement about how 27 = 3 + 24 partitions under F₄-scalar projection. Hierarchy is handled by Z₃ pinning at tau₀ rather than superpartner cancellation; MSUSY = 3500 GeV follows from the REWSB fixed point. Bridge Theorem status: 4 structurally proved + 1 numerical residual (RGE preservation at tan beta = kW = 30 to 0. 25%, pending 2-loop algebraic upgrade). Further corroboration: the framework-internal identity kH = Nc - 1, derived from kₛ = (Nc-1) (Nc+1) = kH x kgrav = 2 x 4 = 8, provides an independent route to kH = 2 that does not invoke the DV-T algebra at all. Step 6 - Cosmology from the same fixed point. Inflation observables (nₛ, r, Nₑ, Aₛ) come from Peter-Weyl mode sums at tau₀; the canonical inflaton is identified as the Goldstone direction of broken no-scale SU (1, 1) /U (1) invariance, phi = sqrt (Nc) * ln (Im tau / Im tau₀). The cosmological constant Lambda / MP⁴ = 2. 827 x 10^-122 follows from Vₙp = (1/2) (kGUT² / Nc^ (5/2) ) |q₀|⁵2 (measured: 2. 850 x 10^-122). H₀ = 67. 26 km/s/Mpc from the same calculation. THE EQUATION tau₀² + tau₀ + 1 = 0 -> tau₀ = e^ (2 pi i / 3) = -1/2 + (sqrt (3) /2) i This is the CM point of the elliptic curve y² = x³ + 1. Every result in the paper flows from modular forms and their derivatives evaluated at this single point. Gauss's h (-3) = 1 pins tau₀ as the unique such point in the Z₃-symmetric class; GVW minimization confirms it as the global vacuum (five independent routes). PREDICTIONS - 26 RESULTS, ZERO FREE PARAMETERS Higgs boson mₕ = 125. 19 GeV | PDG 2023: 125. 20 +/- 0. 11 GeV | 0. 008% (0. 06 sigma; consistent well within 2-loop MSSM theory uncertainty ~+/- 1 GeV) Gauge sector alphaₛ = 1/kₛ = 1/8 | alphaW = 1/kW = 1/30 | alphaGUT = 1/kGUT = 1/26 alphaEM^-1 (0): kEM = 137 (integer geometric prediction) | PDG: 137. 036 | 0. 03% 2-loop MSSM RGE output: 137. 385, within 0. 25% threshold uncertainty sin² thetaW (MGUT) = Nc / (2 Nc + 2) = 3/8 | exact SU (5) tree, standard Georgi-Quinn-Weinberg 1974 value re-expressed in framework integers theta-bar (strong-CP angle) = 0 exactly | current bound |theta-bar| 73 numerical checks in the companion document (v175+, April 2026). STATISTICAL SIGNIFICANCE Twenty-six predictions from zero free parameters. Under standard Bayesian model comparison (Trotta 2008 arXiv: 0803. 4089; Fowlie 2024 arXiv: 2401. 11710), the evidence against the null hypothesis is reported at two distinct levels of independence. Per-observable picture (companion P. 82. 22). Combining all 26 prediction likelihoods against their per-observable priors (product of prior range / precision across observables): Full independent count: log₁0 B ~ 83 (10⁸3 t