Distinction Algebra: From Distinction to Quantum Mechanics, Gravity, and the Standard Model

Distinction Algebra v11.2 (vs v11.1) 1.Four tier promotions [A/identification] / [B/structural] → [A]: §3.1b N_c = 3 (SM anomaly-cancellation 6-condition table, two disjoint forcings); §11.10 ρ = 1 (custodial ℍ_tower SU(2)_L×SU(2)R Theorem 11.10-1, M_W = 79.96 GeV at −0.5 %, S = T = U = 0); §16.12 Page time = ½ t_evap (Hurwitz half identity, conditional on §16.5); §16.17 ξ = 1/6 (CD conformal coupling theorem, (d−2)/(4(d−1))|{d=4}, A_s residual −14.7 % → +0.7 % at N_e ≈ 57.6). 2. Supplement-C absorption: muon g−2, EW precision oblique S/T/U, anomaly cancellation, QEC [7,4,3] Hamming, reheating + ξ + f_NL, de Sitter quantum-gravity content absorbed inline into body §3 / §5 / §10 / §11 / §14 / §16. No separate supplement published. 3. Five new decisive retraction triggers R96–R100: muon g−2 outside m_e-Koide-propagated band; |f_NL| > 1 at >5σ; |ΔN_eff| > 0.1 + oblique zeros; BH-unitarity + CDF-II joint > 3σ; fundamental-constants drift. 4. P85a / R78 retracted: T_c/f_π = √5 sits outside its own pre-registered band [1.5, 2.0]; no §15.1derivation. Superseded by P66/R62 T_c = f_π·√3 at +3.5 % vs HotQCD 156.5 ± 1.5. 5. §17.4 m₂-RGE strict-tier downgrade [A] → [A/B]: eight-step structural chain stays [A], 1-loop DHN numerical landing 4.935 (−1.3 %) sits at [B/derived] pending multi-loop convergence to ≤0.5 % (R85 innerwindow). Presentation correction, not retraction. 6. Λ_QCD canonical-cut reframing per Caswell–Jones–'t Hooft (β₀, β₁ scheme-independent under mass-independent schemes): canonical Λ_QCD(n_f=3) = 332 MeV (0.0 % vs FLAG 332 ± 17); 4-loop MS-bar 327.6 MeV supplementary. §11.2-3a [B/derived] → [B/structural]. 7. §2 axiom-order rewrite: §2.5 retired; §2.1 now presents Axiom N + Axiom D (π_e: Im(𝕆) → ℝ) directly. 8. New §14.13 inflation-closure consolidator (single source of truth for N_e = 57.6, n_s = 0.9653, r = 0.0036, A_s = 2.115×10⁻⁹ via §16.17 ξ = 1/6 closure). 9. Ledger: 73 LIVE / 18 decisive → 77 LIVE / 22 decisive (+5 added R96–R100, −1 retracted R78). Prediction Sheet renumber P82–P86 → P96–P100 to avoid collision with §13 Section A v9.x mass-ratio predictions. 10. No v11.1 headline numerical prediction modified — v11.1 algebraic content preserved verbatim; all v11.2 changes are absorbed numerical witnesses + tier promotions + strict-tier downgrade. --- An algebraic framework deriving the Standard Model and General Relativity from two axioms — Distinction (D) and Norm (N) — plus the Hurwitz / Cayley–Dickson chain ℝ ⊂ ℂ ⊂ ℍ ⊂ 𝕆, with G₂ = Aut(𝕆) as the gauge-content seed. Core architecture. Each normed division algebra plays a forced physical role: - ℝ — classical measurement output - ℂ — quantum-mechanical scaffold (interference) - ℍ — spacetime (signature (1,3) via H₂(ℂ) Pauli identification) - 𝕆 — internal / gauge (SU(3)c via Stab{G₂}(e₁); SU(2)_L × U(1)_Y from the residual G₂ structure) The Single-Scale Theorem reduces the framework to 1 dimensional input + 0 dimensionless inputs generating ~50 catalogued observables. Headline closed-form predictions. - α_EM⁻¹ = 137 = (|W(G₂)|−1)(|W(G₂)|+1) − h(G₂) [A] (+0.026 %, residual closed by 1-loop QED VP at Q* ≈ 3.71 MeV) - α_s(M_Z) = 3√3/44 = γ_Pl²/11 [A] (Theorem G₂-SR: Callias index 8+3+0 = 11 + Killing-form equipartition; +0.16 %) - sin²θ_W = 3/13 [B/structural] (−0.19 %) - N_c = 3 [A] (v11.2 §3.1b two disjoint forcings: Berger–Borel S⁶ stabiliser + SM anomaly cancellation) - N_gen = 3 [A conditional on Axiom 3] from Hurwitz (1898) - ρ = 1 [A] (v11.2 §11.10 custodial ℍ_tower theorem); M_W = 79.96 GeV (−0.5 %); S = T = U = 0 - a_μ^DA = 116 587 440 × 10⁻¹¹ [B/structural] (v11.2 §11.9; ~0.5σ vs PDG, m_e Koide as dominant residual) - m_H = m_t / √2 ~1 % (PT λ=2 + IR fixed point) - M_Pl structural formula: ln(M_Pl/m_t) = (3^(3/4)/2)·[Σ_SM ln(m_3/m_1) + N_B·ln 7], −1.8 % - Λ = 27 · m_ν₁⁴ (+0.82 %) - Λ_QCD(n_f=3) = 332 MeV [B/structural] (v11.2 canonical 2-loop scheme-independent cut; 0.0 % vs FLAG) - Ω_DM / Ω_b = 59/11 [A] (−0.013 % vs Planck 2026) - m_DM/m_p = 5 [A] / [A/B] joint (v11.2 §17.4 G₂ ⊃ SU(3)×SO(4) chiral/pseudo-real RGE chain; structural chain [A], 1-loop DHN landing 4.935 at [B/derived]) - PMNS sin²θ₁₂ / sin²θ₂₃ / sin²θ₁₃ = 4/13, 7/13, 3/137 [A] (Coxeter c³ = −id|_Im ℍ forcing; §13.1) - δ_PMNS = 193.25° [A] (Cartan asymmetry |a₂₁/a₁₂| = 3; §13.8) - Koide K = 2/3 at +0.0009 % via Z₃ + CD equipartition - δ_CKM = arctan √(m_2/m_1) = 65.905° at +0.008 % - ξ = 1/6 [A] (v11.2 §16.17 CD conformal closure; A_s residual −14.7 % → +0.7 % at N_e ≈ 57.6) - Page time t_Page = ½ t_evap [A] (v11.2 §16.12 Hurwitz half identity, conditional on §16.5 BH-CD identification) - f_NL^local ≈ +0.014, f_NL^equil ≈ +0.0023 [B/structural negative] (v11.2 §16.14, both ≪ CMB-S4 σ ≈ 1) - 𝕆 = [7,4,3] perfect Hamming code [A] (v11.2 §3.2c: n=7=dim Im 𝕆, k=4=dim ℍ, d=3=dim Im ℍ=N_c t=1=Axiom-D; C₀ = 1/4 fourfold coincidence) Centerpiece (§17). The Λ⁴(𝕆) integer bridge: γ_Pl² / α_s = 11 = C(8,4) · Ω_b / Ω_m ties the QCD gauge coupling to the cosmological matter budget through pure integer arithmetic of 𝕆. The §17.3 Λ⁴-CS theorem (matter-budget normalizer 70 = 4·11 + 2·13 forced as the unique two-value Coxeter Hodge stratum) closes the LHS at [A]; the §17.4 m₂-RGE theorem closes the RHS structural chain at [A], with the 1-loop DHN numerical landing 4.935 sitting at [B/derived] (joint reading [A/B], v11.2 strict-tier accounting). Visualisation. The Λ²(𝕆) panoramic chart organises 28 SM+GR parameters as 4 concentric hexagons (28 = 4 × 6 + 4) over the SU(3)-stabiliser decomposition; promoted in v11.1 to the k=2 stratum of the Appendix K (𝕆) 128-Room Atlas (Σ C(8,k) = 256 → 128 Hodge-dual-paired rooms, 5-axis address (k, p, G₂ strain, SU(3) sub-rep, (SU(2), Y))). Methodology. All results carry an explicit tier label: [A] (algebraic theorem from D+N), [A/identification] (algebraic + named identification step), [A/B] (joint structural [A] + numerical [B/derived]), [B/structural], [B/derived], [B/identification], [C/open]. A retraction ledger (77 LIVE / 22 decisive triggers) commits the framework to falsifiable predictions tied to upcoming experiments (Hyper-K, DUNE, T2HK, LiteBIRD, HL-LHC, FCC, CMB-S4, DESI DR3, MEG II, atomic-clock fundamental-constants). Reproducibility. 139 standalone Python scripts (numerical witnesses for every quantitative claim) ship alongside the paper, including 235_ckm_multifano_steiner.py (§7.5 audit item (b) [CLOSED — ruled out by exhaustion] via 294-convention + heptagonal Steiner-triple multi-Fano-line scan), 207_pmns_exhaustive.py (18-candidate σ-residual enumeration: DA at 0.4σ, next-best 11.8σ), 218_page_curve.py (Page-curve numerical witness), 219_ew_precision.py (M_W and S/T/U). Companion artefacts. - Full paper PDF + Markdown source - 1-page prediction sheet (PDF + Markdown) - 1-page framework spine (PDF + Markdown) - 33-entry glossary - Λ²(𝕆) panoramic + hexagram charts (PDF + PNG) - 139-script reproducer archive (distinction_algebra_v11_2_scripts.zip) --- License: CC-BY 4.0. ORCID: https://orcid.org/0009-0005-0447-6748.