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

Abstract

Abstract

A derivation of the Standard Model gauge, flavour, Higgs and cosmological sectors from a single algebraic axiom: τ₀² + τ₀ + 1 = 0 — the condition that τ₀ is a primitive cube root of unity, the Z₃ complex-multiplication point of the elliptic curve y² = x³ + 1. Inputs: one structural input (D = 3, three spatial dimensions) and one dimensional anchor (M_Z = 91.1876 GeV, sets the unit of energy). Everything else — gauge couplings, mass ratios, mixing angles, the Higgs mass, the cosmological constant, the Hubble constant — is a prediction. This deposit (Completion Stage V2) supersedes Watford 2026 SM Framework v5 (DOI 10.5281/zenodo.19674289). Reduction-style applications appear in DOI 10.5281/zenodo.19688258 (Li-Liao) and 10.5281/zenodo.19768052 (Riemann Hypothesis dual-proof). Selected Predictions All values predicted from the axiom; PDG / NuFit comparisons use 2024 data. Higgs and Electroweak m_h = 125.193 GeV (PDG 2024: 125.20±0.11; 0.06σ) 1/α_EM(M_Z) = 128.641 (PDG 127.952; 0.54%) sin²θ_W(M_Z) = 0.23058 (0.28%) α_s(M_Z) = 0.11901 (0.86%) CKM and PMNS |V_us| = 1/√20 = 0.22361 (PDG K_ℓ3 0.2233; 0.6σ) — predicts resolution of first-row Cabibbo Anomaly via |V_ud| = 0.97467 γ_UT(CKM) = 65.41° (LHCb ICHEP 2024 64.6°; 0.29σ) δ_CP(PMNS) = 195.633° (NuFit 6.0 within 1σ across readings) sin²θ_13 = 1/45 = 0.02222 (NuFit 0.02195; 0.49σ) sin²θ_12 = 83/270 = 0.30741 (NuFit 0.30714; 0.13%) sin²θ_23 = 17/30 = 0.5667 (NuFit 0.561; 0.50σ) m_ν2/m_ν3 = √3/10 = 0.17321 (NuFit 0.172; 0.70%) Fermion masses m_e/m_τ = |q|^(3/2) = 2.85×10⁻⁴ (PDG 2.88×10⁻⁴; 0.81%) m_μ/m_τ = |q|^(1/2)·(1−N_c/k_W) = 0.0593 (PDG 0.0595; 0.36%) m_c/m_t = 1/360 (PDG 0.00280; 0.71%) m_s/m_b = 3/160 (PDG 0.0189; 0.83%) Cosmology Λ/M_P⁴ = 2.83×10⁻¹²² (Planck 2018 2.85×10⁻¹²²; 0.81%) H_0 = 67.26 km/s/Mpc (Planck CMB 67.4±0.5; 0.21%) — Friedmann equation with G cancelling A_s = 2.116×10⁻⁹ (Planck 2018 2.099×10⁻⁹; 0.58σ) n_s = 29/30 = 0.9667 (Planck 0.9649; 0.42σ) r = 1/300 = 0.00333 (Planck/BICEP-Keck r<0.016; safe) η_B = 5.94×10⁻¹⁰ (PDG 6.10×10⁻¹⁰; 2.6%) N_e = 60 (three-way convergence) Gravity G = 1/4 in Planck units = 1/k_grav (three independent routes: Fubini-Study, BF spectral flow, BF η-invariant) S = A/(4G) — Bekenstein-Hawking factor 4 = |E(F₃)| exact ε_grav = (1−i)/120 (four independent routes) M_P = M_GUT × k_EM^(3/2)/√P = 1.225×10¹⁹ GeV (CODATA 1.221×10¹⁹; 0.37%) Total: 44 predictions framework-internal; 28 under 2%, 32 under 5%, 43 under 10%. Architecture (one paragraph) From τ₀, four independent mathematical streams converge: the nome |q(τ₀)| = exp(−π√3) (fermion mass hierarchies, cosmological constant, primordial amplitudes); Chern-Simons levels k_s = 8, k_W = 30, k_GUT = 26, k_grav = 4 (gauge sector via stabiliser chain SU(3)⊃SU(2)⊃U(1)); the Kähler metric at τ₀ (PMNS angles including δ_CP); APS index theorem on the Hopf bundle (Δ = 1, Ω_m = 1/π). Three independent routes force N_c = 3: Heegner h(−3) = 1 (Gauss 1801), Fradkin's theorem on harmonic-oscillator symmetry, Albert's theorem on exceptional Jordan algebras. What's new vs v5 Anchor count reduced from 5 to 1 (M_Z only) η_B proved as algebraic identity from A_s + framework integers (joins the nome-power family with Λ and A_s) δ_CP and H_0 closed forms incorporated end-to-end All 24 β-function coefficients (1-loop + 2-loop, MSSM + SM) framework-derived from matter content alone (zero imports) m_τ symmetric closed form added Mechanical verification suite (SymPy) caught and corrected one pre-submission arithmetic error Two named bounded technical residuals remain: η_B leptogenesis-chain reduction at script level (corpus-level resolution at archive P.35.40b), and m_τ sub-percent derivation pending y_τ(M_GUT) from A_4 + SU(5) down-type matter coupling (current 8.5% match comparable to other ~5% nome residuals). Honest precision-band note Removing PDG anchors that earlier versions absorbed structural residual through makes the framework's intrinsic 5% nome residual visible in M_P, H_0, M_R1, m_3 (4–15% errors). Predictions independent of M_GUT — lepton mass ratios, m_h via P.82.18, all dimensionless ratios — remain at sub-percent. The drop from earlier "36/44 under 2%" framings to "28/44 under 2%" reflects this honest exposure, not a regression. Deposit Contents File What it is main_paper.md Main paper with end-to-end derivation chain Reviewer_proof_package_combined_v3.md Ten load-bearing proofs with inlined derivations §A.1–§A.22 framework_archive.md Full derivation history and corpus (~14,500 lines, P.1–P.94) Bridge_Theorem_Step3_Proof.md Three-sector closure of Bridge Theorem Step 3 Full_Lagrangian.md Complete action in journal-ready LaTeX, every term tagged Exec_Summary.docx Reviewer-grade executive summary SM_predictions.docx Predictions table ordered by detection difficulty CHANGELOG.md Itemised changes since v5 omega_equation.md One-page distillation five_impossible_things.md Popular-audience essay how_i_built_the_standard_model_extended.docx Methodology narrative (first-person) image1.png, image2.png, image3.png Figures Verification scripts (all run in seconds, no external data): Script What it does framework_formal_verification.py SymPy mechanical proof of 59 arithmetic identities at exact symbolic precision full_derivation_3D_to_SM_completion_stage_v2.py Phase 0 → Phase 5 end-to-end; reports all 44 predictions vs PDG derive_betas_full_2loop.py All 24 β-function coefficients from framework matter content; matches Martin 1997 alpha_1k_geometric_susy.py Standalone framework SUSY-spectrum cross-check Reading order 5-minute first pass: omega_equation.md; run framework_formal_verification.py 30-minute reviewer audit: Exec_Summary.docx → Reviewer_proof_package_combined_v3.md → run all three verification scripts Deep dive: main_paper.md → Bridge_Theorem_Step3_Proof.md → Full_Lagrangian.md → framework_archive.md String-theoretic context The framework's structural elements (T⁶/Z₃ orbifold, GVW superpotential, Magic SUGRA Bridge Theorem, exceptional Jordan algebra h₃(𝕆)) are all string-theoretic. The framework should be understood as a derivation of which compactification describes our universe, framed at the low-energy effective theory level. The Modular Entanglement Fabric is the structural form SUSY takes in this specific vacuum: F-term vanishes at τ₀ (no-scale), Q_α realised geometrically on ℂℙ¹, sparticles structural rather than propagating. String theory provides the UV completion above M_GUT; the framework determines the IR fixed point at τ₀; the MEF describes the symmetry structure of that fixed point. Cross-paper note The "Bridge Theorem (R17)" referenced in the paper (DOI 10.5281/zenodo.19768052) is the same content as archive section P.85 (in-corpus label R14); a cross-paper alias note appears immediately above the P.85 header. arXiv endorsement request If you found this paper to be of merit and would be willing to endorse me to submit an article to the physics.gen-ph section of arXiv, please visit: https://arxiv.org/auth/endorse?x=AGIZZB If that URL does not work, visit http://arxiv.org/auth/endorse.php and enter the following code: Endorsement Code: AGIZZB