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

Distinction Algebra v11.4 (vs v11.3) 1. Dark-sector rebuild. v11.3 outside-dark content preserved verbatim. Three v11.3 claims retracted on C_{G₂}(SU(3)) = {e} (SU(3)_c maximal-rank in G₂): SO(4) dark gauge (§17.2a); ADM n_DM = n_b proof (v11.3 §17.11); m₂-RGE m_DM/m_p = 5 (v11.3 §18.4). Independent kill: Theorem D1 (Frobenius–Schur ε(7) = +1 ⇒ ΔB = 0 on G₂-instanton 't Hooft vertex). 2. Theorem 59/11 — Atlas-PT budget closure (§17.11 / §18.4). dim Λ⁴(𝕆) = 70 splits under SU(3)-branching Sym²₀(7) = 27 into 11 ⊕ 59 by PT-barrier; with 't Hooft 1-loop trace-anomaly factor T(14)/T(7) = 4, three [A] predictions: - Ω_DM/Ω_b = 59/11 = 5.364 (−0.01 % vs Planck 2018) - n_DM/n_b = 4 = C₂(7) = dim ℍ - m_DM/m_p = 59/44 ⇒ m_DM = 1.259 GeV Tree-level PT independently: m_DM/m_p = 3/√5 at 0.054 %. §18.5 integer bridge [A]=[A] (both sides 11, no residual). 3. §17.11.0 dark phenomenology rewritten as gauge-invisible 59-stratum (σ/m ≪ 10⁻³ cm²/g, σ_SI ≪ 10⁻⁴⁸ cm²). §17.11.1: m_DM = 1.259 GeV below WIMP class — sub-GeV nuclear-recoil only (SuperCDMS, NEWS-G, CRESST-III, DarkSide-LowMass 2027+). 4. R-triggers cycled. R49 / R50 / R85 retracted (frozen, no-renumbering); R104–R107 added: m_DM ∈ [1.1, 1.5] GeV, Ω_DM/Ω_b outside 59/11 ± 0.01, DM gauge-singlet, Ω_Λ/Ω_matter ≠ 13/6 at ±5 % CMB-S4. Errata E12 documents the cascade. P91 retired DEAD; P7 recast to Atlas-PT closure; §20.5l new row P86a registers Ω_DM/Ω_b = 59/11 at [A]. 5. (13:5:1) particle-mass narrative retired in favour of Ω_Λ/Ω_matter = 13/6 + Ω_DM/Ω_b = 59/11. New §17.8a: additive Coxeter + imaginary route 13/6 = (h(G₂) + dim Im 𝕆)/h(G₂) at [A], cross-checking §17.8 cyclotomic. Body retraction-residue cleanup across §13.5 / §16.9 / §20.3 / §20.11 / §20.13 / Apx D.7; formal record in §20.4.1 / §20.6 / §20.9 / §A.4 / §Z.6. 6. Audit fixes (2026-05-22). - A — m_t scheme: unified across §19.2 / §11.7 / §14.5 / §14.5-1 / §G.x / §I / Errata E8 on POLE 172.57 GeV (v11.3 mislabel "PDG 2024 MS-bar 172.69"; MS-bar(m_t) ≈ 164.66 rejected at χ² = 3.73). Sub-tenth-percent residuals; R71 (±5 %) unchanged. - B — m_ν₁ label: §14.8 + §20.5d row 46 "exp 0.995 meV" → Λ-formula cross-check (unmeasured; KATRIN ≳ 50 meV). - C — α_s citation: 6 stray cuts unified on PDG 2024 0.1179(9); framework 3√3/44 = 0.118094 and R76 window unchanged. - D — Errata E3 extended: N_e = 57.6 from §17.17 ξ = 1/6 (§14.13 consolidator); near-convergence with retracted 57 is arithmetic coincidence within Starobinsky 50–60. 7. Resolution-doc execution (2026-05-22, clean-body, no new Errata). (i) §2 Premise P retirement: §2.2 / §2.4 / §3.2a [A modulo P] → [A] (δ² = ε excluded by Axiom N productive clause). (ii) §3.2 (iii) U(1)_Y rewrite: dim-counting "12−8−3=1" → ℝ → ℂ CD first-doubling phase + C_{G₂}(SU(3)_c) = {e}; dim 𝔤_SM = 12 = |W(G₂)| as Corollary. (iii) §18.4 Step 3: "Sym², confined" → "⊂ Sym²₀(7) = 27, ⊄ Λ²(7) ⊃ adj G₂ = 14". (iv) §19.6a: "27 of 28 derived" → "27 of 28 algebraically constrained from a single dimensional input". 8. Hygiene: 10 in-body "Honest/Honesty" labels stripped. §11.7 L2638 citation hygiene (3 paragraphs; Theorem G₂-SR derivation at L1449 / L2603–2654 unaffected, [A] preserved). 9. Ledger 80 / 24 → 81 LIVE / 23 decisive. Python 151 = 150 v11.3 base − 3 retracted (165_ADM_5proof.py, 166_omega_partition.py, 202_dark_qcd_rge.py) + 4 new (241_D1_frobenius_schur.py, 242_atlas_PT_budget.py, 243_partition_dual_route.py, run_all_v11_4.py). No tier promotions outside §17.11 / §18.4 / §18.5. --- 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); U(1)_Y from CD first-doubling phase (§3.2 (iii), v11.4) - ℍ — spacetime (signature (1,3) via H₂(ℂ) Pauli identification) - 𝕆 — internal / gauge (SU(3)_c via Stab_{G₂}(Axiom D) [A]; SU(2)_L × U(1)_Y from residual G₂ structure with C_{G₂}(SU(3)_c) = {e} [A]) The Single-Scale Theorem reduces the framework to 1 dimensional + 0 dimensionless + 0 identification (composite [A]) 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 % vs PDG 2024 0.1179(9)) - sin²θ_W = 3/13 [B/structural] (−0.19 %) - SU(3)_c [A] (§3.1b Stab_{G₂}(Axiom D) = SU(3); π_e Riesz uniqueness on Im 𝕆) - SM gauge group SU(3) × SU(2) × U(1) [A] (§3.2 CD-tower ℍ disambig + v11.4 ℝ → ℂ first-doubling for U(1)_Y + centralizer C_{G₂}(SU(3)_c) = {e}) - N_c = 3 [A] (SM anomaly cancellation, two disjoint forcings) - N_gen = 3 [A conditional on Axiom 3] from Hurwitz (1898) - ρ = 1 [A] (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] (~0.5σ vs PDG, m_e Koide as dominant residual) - m_H = m_t / √2 ~1 % (PT λ=2 + IR fixed point); m_t = 172.57 GeV POLE (v11.4 audit-A) - 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 [A] (§18.4; Robertson uncertainty + BPS shape-invariance + Bogomolny δE = 0) - Λ_QCD(n_f = 3) = 332 MeV [B/structural] (canonical 2-loop; 0.0 % vs FLAG) - m_G = √27 · Λ_QCD ≈ 1725 MeV [A] (§15.3-1; lattice 1710 ± 50, +0.9 %) - √σ = 439.2 MeV [A] (§15.4; σ = (7/4)·Λ_QCD² ⇒ experimental 440 MeV at −0.2 %; decisive P101) - Ω_DM / Ω_b = 59/11 ≈ 5.364 [A] (v11.4 Theorem 59/11, Atlas-PT budget closure; −0.01 % vs Planck 2018) - n_DM / n_b = 4 [A] (v11.4) - m_DM = 1.259 GeV [A] (v11.4 m_DM/m_p = 59/44; tree-level cross-check 3/√5 at 0.054 %; below WIMP class, sub-GeV nuclear-recoil channel only) - Ω_Λ / Ω_matter = 13/6 [A] (v11.4 §17.8a additive Coxeter + imaginary route, cross-checking §17.8 cyclotomic) - PMNS sin²θ₁₂ / sin²θ₂₃ / sin²θ₁₃ = 4/13, 7/13, 3/137 [A] (§3.1c cascade from SU(3)_c; long-root A₂ / short-root forced; 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] (§17.17 CD conformal closure; A_s residual −14.7 % → +0.7 % at N_e ≈ 57.6) - Page time t_Page = ½ t_evap [A] (§17.12 Hurwitz half identity, conditional on §17.5 BH-CD identification) - f_NL^local ≈ +0.014, f_NL^equil ≈ +0.0023 [B/structural negative] (§17.14, both ≪ CMB-S4 σ ≈ 1) - 𝕆 = [7,4,3] perfect Hamming code [A] (§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) - G₂ gauge fixing (1-bit pin) [A] (§3.4; Berger 1955 S⁶-transitivity, structurally irreducible) - φ-isomorphism Theorem 2.8-1 [A] (CD formula unique up to G₂ gauge — std/alt products bridge via ℤ₂ relabelling) - Robertson uncertainty as theorem [A] (§2.9; Cauchy–Schwarz on ℂ ⊗ ℍ Hilbert space, not postulate) Centerpiece (§18). 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 §18.3 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]. In v11.4, the §18.4 closure is rebuilt: the v11.3 m₂-RGE chain (which forced m_DM/m_p = 5) is retracted on C_{G₂}(SU(3)) = {e} and replaced by Theorem 59/11 — Atlas-PT budget closure, where the SU(3)-branching Sym²₀(7) = 27 with PT barrier active/inactive splits dim Λ⁴(𝕆) = 70 into 11 ⊕ 59, giving Ω_DM/Ω_b = 59/11 at [A] without dark-sector input. §18.5 integer bridge [A]=[A] (both sides 11, no residual). §15 mass-gap centerpiece. Yang–Mills mass-gap closure for N = 3 on ℝ³·¹ × S⁶. Theorem 15.3-1 (BPS + Peter–Weyl + subsector inequality Δ₄ ≥ Δ₁₀ > 0) delivers a strictly positive spectral gap; Theorem 15.3-2 (Osterwalder–Seiler lattice OS + Prokhorov tightness + OS 1975 Thm 3.2) reconstructs Wightman axioms in 4D. Three ingredients from D + N: Axiom D supplies the singlet projection; N → 𝕆 → G₂ → S⁶ = G₂/SU(3) supplies the compact coset (hence Peter–Weyl); N → 𝕆 → G₂ → HILP/BILL kink supplies the BPS structure (hence Pöschl–Teller shape invariance). Numerical readings: m_G = √27 · Λ_QCD ≈ 1725 MeV at +0.9 % vs lattice 1710 ± 50; √σ = 439.2 MeV at −0.2 % vs experimental 440 (decisive R101). 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 (81 LIVE / 23 decisive triggers, v11.4) 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, lattice QCD string-tension precision, sub-GeV nuclear-recoil DM searches from 2027+). Reproducibility. 151 standalone Python scripts (numerical witnesses for every quantitative claim) ship alongside the paper, including the v11.4 dark-sector rebuild bundle (241_D1_frobenius_schur.py Frobenius–Schur indicator ε(7) = +1; 242_atlas_PT_budget.py Λ⁴(𝕆) splitting 70 = 11 ⊕ 59; 243_partition_dual_route.py tree-level m_DM/m_p = 3/√5 cross-check; run_all_v11_4.py full v11.4 sweep), preserving the v11.3 Yang–Mills bundle (236_string_tension.py σ = (7/4)·Λ_QC