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 Nc = 3 (SM anomaly-cancellation 6-condition table, two disjoint forcings) ; §11. 10 ρ = 1 (custodial ℍₜower SU (2) L×SU (2) R Theorem 11. 10-1, MW = 79. 96 GeV at −0. 5 %, S = T = U = 0) ; §16. 12 Page time = ½ tₑvap (Hurwitz half identity, conditional on §16. 5) ; §16. 17 ξ = 1/6 (CD conformal coupling theorem, (d−2) / (4 (d−1) ) |d=4, Aₛ residual −14. 7 % → +0. 7 % at Nₑ ≈ 57. 6). 2. Supplement-C absorption: muon g−2, EW precision oblique S/T/U, anomaly cancellation, QEC 7, 4, 3 Hamming, reheating + ξ + fNL, 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 mₑ-Koide-propagated band; |fNL| > 1 at >5σ; |ΔNₑff| > 0. 1 + oblique zeros; BH-unitarity + CDF-II joint > 3σ; fundamental-constants drift. 4. P85a / R78 retracted: Tc/f_π = √5 sits outside its own pre-registered band 1. 5, 2. 0; no §15. 1derivation. Superseded by P66/R62 Tc = f_π·√3 at +3. 5 % vs HotQCD 156. 5 ± 1. 5. 5.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. ΛQCD canonical-cut reframing per Caswell–Jones–'t Hooft (β₀, β₁ scheme-independent under mass-independent schemes): canonical ΛQCD (nf=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 (πₑ: Im (𝕆) → ℝ) directly. 8. New §14. 13 inflation-closure consolidator (single source of truth for Nₑ = 57. 6, nₛ = 0. 9653, r = 0. 0036, Aₛ = 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 StabG₂ (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. - αEM⁻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) - αₛ (MZ) = 3√3/44 = γPl²/11 A (Theorem G₂-SR: Callias index 8+3+0 = 11 + Killing-form equipartition; +0. 16 %) - sin²θW = 3/13 B/structural (−0. 19 %) - Nc = 3 A (v11. 2 §3. 1b two disjoint forcings: Berger–Borel S⁶ stabiliser + SM anomaly cancellation) - Ngen = 3 A conditional on Axiom 3 from Hurwitz (1898) - ρ = 1 A (v11. 2 §11. 10 custodial ℍₜower theorem) ; MW = 79. 96 GeV (−0. 5 %) ; S = T = U = 0 - a_μDA = 116 587 440 × 10⁻¹¹ B/structural (v11. 2 §11. 9; ~0. 5σ vs PDG, mₑ Koide as dominant residual) - mH = mₜ / √2 ~1 % (PT λ=2 + IR fixed point) - MPl structural formula: ln (MPl/mₜ) = (3^ (3/4) /2) ·ΣSM ln (m₃/m₁) + NB·ln 7, −1. 8 % - Λ = 27 · m_ν₁⁴ (+0. 82 %) - ΛQCD (nf=3) = 332 MeV B/structural (v11. 2 canonical 2-loop scheme-independent cut; 0. 0 % vs FLAG) - ΩDM / Ωb = 59/11 A (−0. 013 % vs Planck 2026) - mDM/mₚ = 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) - δPMNS = 193. 25° A (Cartan asymmetry |a₂₁/a₁₂| = 3; §13. 8) - Koide K = 2/3 at +0. 0009 % via Z₃ + CD equipartition - δCKM = arctan √ (m₂/m₁) = 65. 905° at +0. 008 % - ξ = 1/6 A (v11. 2 §16. 17 CD conformal closure; Aₛ residual −14. 7 % → +0. 7 % at Nₑ ≈ 57. 6) - Page time tPage = ½ tₑvap A (v11. 2 §16. 12 Hurwitz half identity, conditional on §16. 5 BH-CD identification) - fNLˡocal ≈ +0. 014, fNLᵉquil ≈ +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=Nc t=1=Axiom-D; C₀ = 1/4 fourfold coincidence) Centerpiece (§17). The Λ⁴ (𝕆) integer bridge: γPl² / αₛ = 11 = C (8, 4) · Ωb / Ωₘ 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 235ckmₘultifanoₛteiner. py (§7. 5 audit item (b) CLOSED — ruled out by exhaustion via 294-convention + heptagonal Steiner-triple multi-Fano-line scan), 207ₚmnsₑxhaustive. py (18-candidate σ-residual enumeration: DA at 0. 4σ, next-best 11. 8σ), 218ₚagecurve. py (Page-curve numerical witness), 219ₑwₚrecision. py (MW 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 (distinctionₐlgebraᵥ11₂ₛcripts. zip) --- License: CC-BY 4. 0. ORCID: https: //orcid. org/0009-0005-0447-6748.