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

Distinction Algebra v11. 6 (vs v11. 5) 1. Koide + Majorana-kernel strengthening; v11. 5 algebraic content and ledger structure preserved. The headline change is a tier refinement of the lepton-sector relation and an honesty pass on the dark-sector / baryogenesis tiers — no new free parameters, no change to any numerical prediction. 2. Koide K = 2/3: B/structural → A/structural (§19. 4). The value is fixed by unbiased per-irrep counting of the ℤ₃ distinctions (r² = 2 = balance midpoint between democracy K = 1/3 and dominance K = 1; ‖v_ℝ‖² = ‖v_ℂ‖²), with the G₂ ℤ₃-symmetric Higgs giving the closed form K = (h−2) /h = 2/3. The "structural" suffix is retained deliberately: the octonionic generation mass matrix is degenerate (H|gen ∝ I₃ for every Higgs direction, companion script 255b), so the algebra is silent on the weighting; per-irrep equipartition is the unique well-defined zero-information reading of that silence (the faithful ℤ₃ rep is ℝ-irreducible, so per-dimension weighting is not well-defined), not a value the algebra forces. Residual +0. 0009 % = O (y_τ²). 3. §11. 2 ΛQCD (nf = 3): B/structural → A. The dimensionless ratio Λ/MZ = F (αₛ, β₀, β₁) is scheme-invariant at the 2-loop Caswell–Jones–'t Hooft boundary; the absolute 332 MeV follows from measured MZ and thresholds. 4. §14. 8 MR: the Majorana Kernel Identity Kᵢj = Σc φ₁₈₂ φ₁₉₂ = I₆ replaces the per-channel NDA estimate (proof quality; tier A unchanged; Tr (K) = h = 6 → e^ (−6) ). 5. §7. 7 κ, BWilson held at A/identification (honesty correction). The values κ = 3^ (3/4) /√2, B = 3^ (5/4) are reconfirmed A-valued, but the overall tier remains A/identification — the KK Wilson-line per-generator normalisation is a named identification step; the earlier "→ A, residue closed in §B/M1" reading is withdrawn (§B carries no such closure). 6. §14. 10 ΔNₑff = 0 → A/structural negative (per-type exclusion, each binding constraint A). 7. Dark-sector tier honesty pass. The dark-sector predictions ΩDM/Ωb = 59/11, nDM/nb = 4, mDM = 1. 259 GeV — and the §18. 5 integer-bridge cosmology side — are relabelled A → A/structural. The 11/59 dimensional split (rep theory + Pöschl–Teller barrier, Ω-independent) is A; mapping it to the energy ratio assumes per-cell equipartition over the reducible Λ⁴ (𝕆) — the same counting principle as the §19. 4 Koide value, parameter-free but not forced by the algebra alone — and the "Λ⁴ = matter budget" identification has no k=4 bijection theorem (unlike the k=2 Theorem 9. 0-1 fixing the 28 SM parameters). This is an honest tier revision, not a retraction: the predictions and their falsifiers (R104–R107) are unchanged. 8. §17. 10 ηB baryogenesis reframed to its structural determination. ηB is fixed structurally, not by thermal leptogenesis: dynamics + sign (ηB > 0, matter domination) are A via the SO (8) triality cascade S₃ → ℤ₂ → 1 (§9. 7. 3, Axiom D/N break 8ᵥ / 8ₛ / 8c) ; magnitude is A/structural as the baryon density ηB = 2. 74×10⁻⁸ Ωb h² = 6. 37×10⁻¹⁰ with Ωb = (11/70) (6/19) (§18 integer bridge; absolute value carries H₀ B/derived). The §17. 10 thermal-leptogenesis chain is the Standard-Model rendering of the same baryon number; its thermal realisation is non-viable (MR = 3×10¹⁶ ≫ TRH = 2. 3×10⁹; SU (3) -singlet νR massless at leading order), documented as a quantified no-go — expected for a structural framework, and not affecting the observable. 9. §18. 3 Theorem Λ⁴-CS correction. The "uniqueness" claim was sharpened: seven of the nine Hodge strata have exactly two distinct Coxeter eigenspace dimensions, so the feature unique to k=4 is that its pair is specifically 11, 13 (the gauge denominators), not "having two values. " The partition 70 = 4·11 + 2·13 and the selection of k=4 (the unique Hodge-self-dual central stratum carrying the trace-anomaly 27 = Sym²₀ (7) ) are both forced; the single post-hoc element is the cross-sector coincidence that the forced integers 11, 13 equal the gauge denominators. 10. §16. 4 PP Type E fK/f_π recast as NLO ChPT B/structural. The mis-stated single-power value was corrected (the bare (4/3) ^ (2/7) = 1. 086 undershoots by 9 %), then rebuilt on the correct NLO form — DA chiral logs (1. 064) + the O (p⁴) constant L₅ = Nc/ (16π² (dim ℍ) ²) = 3/ (256π²) from DA scalar-resonance saturation (MS = dim ℍ · Λ) — giving fK/f_π = 1. 1936 (+0. 02 %) with zero external numbers. 11. Cell-view clean recomposition (§19. 3 / §19. 6a / §21. 1). The 28-cell Λ² (𝕆) scorecard is aligned to the Theorem 9. 0-1 partition of the genuine 28 SM + Majorana-neutrino parameters: 10 A (3 gauge + the three PMNS angles + δPMNS + the two Majorana phases + the Higgs VEV v) + 1 A/B (mH) + 1 A/structural (θ̄, strong-CP) + 16 B/structural (4 CKM + 9 charged-fermion masses + 3 neutrino masses). Derived companions (Koide K, JPMNS, λH) are reported at their tiers but not counted as separate cells; the gravity/cosmology quantities MPl and Λ are not among the 28 — MPl is the single dimensional input (mₜ ≡ MPl) and Λ is a prediction (Λ = 27 m_ν₁⁴), departing from the PDG/Quigg SM+GR basis where Λ is a free parameter. 12. Numerical / citation refresh. αₛ unified on PDG 2024 0. 1179 (9) (framework 3√3/44 = 0. 118094 at +0. 16 %, +0. 22σ, inside the 2σ band) ; mₜ pole unified on 172. 57 GeV; Koide experimental residual sign unified at +0. 0009 % under the (framework − obs) /obs convention; Pöschl–Teller / Cayley–Dickson en-dash and CODATA digit normalisation. Errata (Apx Z. 6) condensed to ≤ 100 words per entry; full-document review-pass corrections (TBM eigenvector signs, Casimir index/geometric normalisation note, stale cross-references). --- 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) ) - ℍ — spacetime (signature (1, 3) via H₂ (ℂ) Pauli identification) - 𝕆 — internal / gauge (SU (3) c via Stab₆䃒 (Axiom D) A; SU (2) L × U (1) Y from residual G₂ structure with C₆䃒 (SU (3) c) = e A) The Single-Scale Theorem reduces the framework to 1 dimensional + 0 dimensionless + 0 identification input (composite A) generating ~50 catalogued observables. The 28 SM + Majorana-neutrino parameters are read as the 28 cells of Λ² (𝕆) (Theorem 9. 0-1 bijection, p < 10⁻⁶ against accidental survival). Headline closed-form predictions (tier in brackets; honest tier accounting throughout). - 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 B/derived) - αₛ (MZ) = 3√3/44 = γPl²/11 A (Theorem G₂-SR: Callias index 8+3+0 = 11 + Killing-form equipartition; +0. 16 % / +0. 22σ vs PDG 2024 0. 1179 (9) ) - sin²θW = deg (φ) /Φ₆ (dim ℍ) = 3/13 A (−0. 19 % residual = MW → MZ 1-loop running, scheme convention) - SU (3) c A (§3. 1b Stab₆䃒 (Axiom D) = SU (3) ; Riesz uniqueness on Im 𝕆) ; SM gauge group SU (3) × SU (2) × U (1) A; Nc = 3 A; Ngen = 3 A conditional on the minimality assumption from Hurwitz (1898) - ρ = 1 A (custodial ℍₜower theorem) ; MW = 79. 96 GeV (−0. 5 %) ; S = T = U = 0 - Spin-2 self-coupling → GR uniqueness A (Deser–Wald via G₂-invariant 3-form uniqueness on Im 𝕆) - a_μDA = 116 587 440 × 10⁻¹¹ B/structural (mₑ Koide the dominant residual) - mH = mₜ/√2 (A/B: framework identity A, −2. 5 % residual at B scheme-closable to ~0. 5 % at the MS-bar top scale) ; λH = 1/8 = mH²/2v² (derived companion) ; v = mₜ√2 = MPl · αₛ¹8 A; mₜ = 172. 57 GeV pole (single dimensional input) - MPl structural formula: ln (MPl/mₜ) = (3^ (3/4) /2) · ΣSM ln (m₃/m₁) + NB · ln 7, −1. 8 % - MR = MPl · e^ (−h (G₂) ) = MPl · e^ (−6) A (Theorem 14. 8-1 Majorana Schur factorization; Kernel Identity K = I₆) ; m_ν₁ = mₜ²/MR A; ν PT-barrier coefficients 7/9, 7/ (6π) A; G₂-equivariant Yukawa (FH ODE n = 3 unique) A - Λ = 27 · m_ν₁⁴ A (§17. 4; Robertson uncertainty + BPS shape-invariance) ; Λ is predicted, not a free SM+GR parameter - ΛQCD (nf = 3) = 332 MeV (scheme-invariant ratio Λ/MZ A, §11. 2; absolute carries measured MZ, B/structural; 0. 0 % vs FLAG) ; mG = √27 · ΛQCD ≈ 1725 MeV A (lattice 1710 ± 50, +0. 9 %) ; √σ = 439. 2 MeV A (σ = (7/4) ·ΛQCD²; experimental 440 at −0. 2 %; decisive P101) - ΩDM/Ωb = 59/11 ≈ 5. 364 A/structural (Theorem 59/11, Atlas-PT budget; dimensional split A, energy-ratio mapping A/structural; −0. 01 % vs Planck 2018) ; nDM/nb = 4 A/structural; mDM = 1. 259 GeV A/structural (mDM/mₚ = 59/44; gauge-singlet, sub-GeV nuclear-recoil channel only) - Ω_Λ/Ωₘatter = 13/6 A (§17. 8a additive Coxeter-data partition, independent of the dark sector; −0. 4 % vs Planck 2018) - ηB = 6. 37 × 10⁻¹⁰: dynamics + sign A (SO (8) triality cascade §9. 7. 3) ; magnitude A/structural (baryon density, Ωb = (11/70) (6/19), §18; absolute value carries H₀ B/derived) ; +1–4 % vs Planck 2018 + BBN 6. 12 (4) × 10⁻¹⁰ - PMNS sin²θ₁₂ / sin²θ₂₃ / sin²θ₁₃ = 4/13, 7/13, 3/137 A (§3. 1c cascade from SU (3) c; Coxeter c³ = −id|Im ℍ forcing, §13. 1) ; δPMNS = 193. 25° A (Cartan asymmetry, §13. 8) ; Majorana phases α₂₁ = α₃₁ = 0 A - θ̄ = 0 A/structural (Strong CP; 𝔽₂ = GF (4) rank-1 + Fano ℤ₃ — no second CP phase to populate the QCD vacuum) - ΔNₑff = 0 A/structural negative (§14. 10, per-type exclusion) - Koide K = 2/3 A/structural (§19. 4; ℤ₃ + CD equipartition on the proven-degenerate generation algebra; +0. 0009 %) ; δCKM = arctan√ (m₂/m₁) = 65. 905° B/structural (+0. 008 %) - ξ = 1/6 A (§17. 17 CD-conformal closure) ; Starobinsky inflation A (circularity-exclusion theorem) ; nₛ / r / Aₛ observables A (propagation through §14. 13 / §17. 17 / §17. 9) - Page time tPage = ½ tₑvap A (§17. 12 Hurwitz half identity, conditional on §17. 5