The Standard Model from a CM Point and the Exceptional Jordan Algebra

Authors: Watford, Paul Version: V102 (2026-05-30) — minimal publish. Two papers + two scripts. ## What this is A deterministic computational framework that takes one polynomial axiom (Φ₃(τ) = 0) and one dimensional anchor (M_Z = 91.1876 GeV) and outputs the Standard Model, cosmology, and hadron spectroscopy in closed form at sub-σ to sub-permille precision against current PDG and Planck data. Two parameters, different in kind: - **Φ₃(τ) = τ² + τ + 1 = 0** — a mathematical constraint, no fitted number. - **M_Z = 91.1876 GeV** — the only empirical input, setting the absolute scale. Everything else is derived: gauge couplings, the gravitational coupling, all nine charged-fermion masses, neutrino masses, CKM and PMNS angles, cosmology parameters, the QCD scale, the proton mass, hadron spectroscopy from light pseudoscalars through bottomonium, the LFH soft-wall confinement scale, the EW vacuum expectation value, the Higgs mass, and the GUT scale itself. ## What this isn't I have not invented new mathematics. I have bolted together a thousand years of other people's work — cyclotomic fields, modular forms, exceptional Jordan algebras, L-functions, supersingular reduction, motivic periods, no-scale supergravity, light-front holography — into a single derivation chain. The mathematics belongs to Gauss, Heegner, Albert, Chevalley–Schafer, Deuring, Mihailescu, Beilinson, Deligne, Witten, Bär, Cremmer, Verlinde, Dubois-Violette–Todorov, Feruglio, Giudice–Masiero, de Alfaro–Fubini–Furlan, Brodsky–de Téramond–Dosch, and many others. The chain that connects them is mine. The framework's distinguishing claim is narrow: the specific joint identification of these tools, applied at the two complex-multiplication points τ₀ = e^(2πi/3) and τ₁ = i, fixes every free parameter of the Standard Model. That is the falsifiable content. ## What V102 changes versus V101 V102 closes three structural gaps that V101 left open, all via the May 2026 KT-RG analysis. The framework's predictive content is unchanged; the foundation under it is significantly firmer. **M_GUT closed in framework primitives (§5.2.1 main).** The previous statement "every Standard Model dimensional input except M_Z is derivable" was conditional on M_GUT being externally supplied. The closure **M_GUT = M_Z · |q(τ₀)|^(−N_c·k_H) = M_Z · e^(6π√3) = 1.377 × 10¹⁶ GeV** removes that conditionality. The integer 6 = N_c·k_H is the order of τ₀'s stabiliser in SL₂(ℤ), the order of the Eisenstein unit group ℤ[ω]^×, and lcm(stab_SL₂(τ₀), stab_SL₂(τ₁))/2 — four independent readings at N_c = 3. Within the GUT window, n = 6 is uniquely forced: the adjacent integer factor is e^(π√3) ≈ 230.8, so n = 5 undershoots by 200× and n = 7 overshoots by 200×. **The framework's scheme is on-shell, not by choice but by structure (KT-RG, May 2026).** Every closed-form physical prediction of the framework is either a pole quantity (pole masses, pole-mass ratios) or a coupling at a specific physical scale (α_GUT⁻¹ at M_GUT, α_EM⁻¹ at low energy). Not one is a generic MSbar quantity. R.16 — the W-Z mass ratio (M_W/M_Z)² = 7/9 — is therefore on-shell by construction (mass ratios live in the on-shell scheme by definition), and the framework's M_GUT is a modular scale, not the MSbar coupling-meeting scale of MSSM literature. The earlier "5% gap" between framework M_GUT = 1.377 × 10¹⁶ and the literature 1.45 × 10¹⁶ was scheme-mixing — analogous to the Δr correction in EW. With consistent on-shell bookkeeping throughout, the fully-geometric chain M_Z → M_GUT → M_SUSY → v → m_h closes at +0.21% absolute, with the entire residue attributable to known physics (M_W tension + standard MSSM RGE threshold). **The Clebsch-O(1) test selects n = 6 uniquely at the bottom of the ladder (companion §15.5).** The B–L flat direction protected by the exceptional cubic norm is lifted at order n by W = λΦⁿ/M_Pl^(n−3), giving v_R ~ (M_SUSY · M_Pl^(n−3))^(1/(n−2)). Inverting the seesaw with the observed atmospheric neutrino mass, M_SUSY fixed by the framework, m_D ~ m_t in SO(10), and demanding the Clebsch g to be O(1) (which Clebsch–Gordan coefficients are by representation theory), only n = 6 survives: n = 5 needs g ~ 7, n = 6 needs g ~ 0.4 (✓ O(1)), n = 7 needs g ~ 0.06. The same integer 6 = N_c·k_H closes both ends of the mass ladder — the climb M_Z → M_GUT through six nome powers, and the descent through a dimension-6 flat-direction operator to v_R and m_ν. **V101 inheritances remain valid and unchanged:** - Chain 7 closed. EW VEV v = 246.22 GeV derived analytically from the strong scale via REWSB; m_h matches PDG at 0.004%. - Hubble tension resolved with two anchors. τ₀ predicts H₀ = 67.26 km/s/Mpc (Planck-side, +0.06σ); τ₁ predicts H₀ = 73.50 km/s/Mpc (local-side, +0.22σ to Casertano 2026 H0DN). Two ladders measuring the same theory at the two CM anchors. - Cosmological constant, baryon asymmetry, and leptogenesis closed via modular-invariance preservation + the Sakharov-cyclotomic identity. - The neutrino sector is octonionic: Majorana channel = forced E₆ invariant 27·27·351′ (companion §15.1), M_R eigenvalue pattern 2:2:1 from Y(τ₀) (§15.2), and the B–L scale is a flat direction protected by the cubic norm I₃ of h₃(𝕆) — the freedom is a consequence of the exceptional structure, not a gap (§15.4). **Honest framing of m_h precision (geometric_SUSY.py three-tier output):** - **Tier R:** m_h/v = 0.5085 from the master cyclotomic formula — pure framework primitives, no dimensional input, scheme-free, exact. - **Tier 2:** m_h = 125.46 GeV from M_Z alone via the fully-geometric chain — +0.21% from PDG, +2.4σ. All residue is known physics (M_W tension + MSSM RGE threshold), not framework error. - **Tier 3:** m_h = 125.2044 GeV at 0.04σ when measured v is used as the second anchor (master cyclotomic + measured v). ## The framework as a runnable circuit - **2 inputs:** the axiom and the dimensional scale. Only the second is empirical. - **13 hubs:** each containing one irreducible mathematical fact (master cyclotomic theorem, Chowla–Selberg evaluation, Albert algebra h₃(𝕆), Klein's quotient A₄ ≅ PSL₂(ℤ)/Γ̄(3), E₈ ⊃ E₆ × SU(3)_F, SO(8) triality, and seven others). - **64 gates** in nine classes: AND, OR, XOR, NOT, MIX, SPLIT, CASCADE, SELECT, TRIALITY. - **11 derived integers:** {k_H, k_grav, k_s, k_W, k_GUT, α_GUT⁻¹, k_EM, Φ_3, Φ_4, Φ_6, Φ_12} = {2, 4, 8, 30, 26, 27, 137, 13, 10, 7, 73} at N = 3. - **2 hidden cyclotomic primitives:** Φ₄(k_H) = 5 (τ₀-side) and Φ₄(k_grav) = Φ₈(k_H) = 17 (τ₁-side; dual via the level-4-at-k_grav equals level-8-at-k_H identity at k_H = 2). - **1 structural identity:** R_conf = Φ₄(k_H)²/(k_H·Φ_6) = 25/14 — framework analog of dAFF uniqueness. - **7 polynomial identities** of the form (N − 3)·Q_i(N), simultaneously vanishing only at N = 3. - **57 observables** computed in framework.py (with closed forms, PDG comparison, sigma deviations), spanning gauge couplings, charged-fermion masses, neutrino sector, CKM/PMNS, cosmology, hadron spectroscopy, and the EW sector. Many additional closed-form parameterisations exist as cross-checks across the V14.7 inventory. Run `python3 framework.py` and the verification cascade executes. Press 0 at the menu for full verification with PDG comparison. Run `python3 geometric_SUSY.py` and the Higgs-mass calculation runs. Press 0 for the loop-ladder + tier discipline output; press 5 for the fully-geometric M_Z → m_h chain in isolation. ## What's in this deposit ``` The Standard Model from a CM Point Main.md 28,243 words · §3 Higgs mass via master cyclotomic formula · §4 Fermion sector closures · §5 Cross-sector consistency (REWSB, M_GUT closure, M_W) · §5.2.1 The GUT scale closure (new in V102) · §6 Cosmology + gravity sector · §7-9 Strong CP, inflation, off-anchor closures The Standard Model from a CM Point Companion.md 10,069 words · §1 The closed-form observables inventory · §3 Two-anchor Hubble tension · §8 Gravity sector extended content · §10.5 The M_GUT closure · §15 The neutrino sector — octonionic seesaw (new in V102) · §16 Three algebraic levels of gravity framework.py 1,597 lines · 57 observables, full PDG comparison · 11-integer ladder + 2 hidden cyclotomic primitives · M_GUT closure as O.56, m_h/v as O.57 · Interactive menu, press 0 to run geometric_SUSY.py ~900 lines · Master cyclotomic m_h proof (no loop input) · Fully-geometric M_Z → m_h chain (Tier 2) · Three-tier honest framing (R / 2 / 3) · KT-RG scheme analysis documented inline · Interactive menu, press 0 to run, press 5 for the fully-geometric chain ``` ## Selected predictions All predictions use zero continuous free parameters beyond the structural input (Φ₃(τ) = 0) and the dimensional anchor (M_Z). Every value below is reproduced byte-for-byte by the deposited scripts. Observable | Closed form | Value | Measured | Match ---|---|---|---|--- sin²θ_W(M_GUT) | N/(2·k_grav) = 3/8 | 0.375 | 0.375 | exact α_GUT⁻¹ | N³ = dim h₃(𝕆) | 27 | ~27 | exact α_s(M_Z) | k_grav·Φ_6/(k_EM·√3) = 28/(137√3) | 0.1180 | 0.1179(10) | +0.10σ m_h | master cyclotomic: √(2·v²·bracket) | 125.204 GeV | 125.20(11) | +0.04σ m_h/v | √(2·bracket), pure primitives | 0.5085 | 0.5085 | scheme-free v (EW VEV) | from M_Z via REWSB (§5.1) | 246.22 GeV | 246.22(1) | <1σ M_GUT | M_Z·e^(6π√3) (§5.2.1, modular) | 1.377×10¹⁶ GeV | — | framework-native m_t (pole) | M_Z·91/(34√2) | 172.58 | 172.57(29) | +0.02σ m_b (MSbar) | Λ_QCD·289√3/26 | 4.180 | 4.18(3) | −0.00σ m_τ (pole) | M_Z·5/(26π²) | 1.77678 | 1.77686(0.00012) | 0.005% m_p/m_e | 4·k_grav·α_GUT⁻¹·Φ₄(k_grav) = 4·27·17 | 1836 | 1836.153 | 8×10⁻⁵ Λ_QCD⁽⁵⁾ | M_Z/(k_grav·N·Φ₄(k_H)·Φ_6) = M_Z/420 | 0.2171 GeV | 0.210(14) | +0.51σ m_p | M_Z·13/1260 | 940.82 MeV | 938.27 MeV | +0.27% m_π⁰ | M_Z/k_GUT² = M_Z/676 | 134.89 MeV | 134.977 MeV | −0.06% f_π | Λ_QCD·