================================================================================ QUANTITATIVE DATA FROM PRIOR ART — 24-CELL PHYSICS ================================================================================ Date: 2026-03-21 Purpose: Collect specific numbers from published 24-cell/D4/F4 physics work for comparison with our framework and engine data. ================================================================================ ================================================================================ 1. ALI (2025) — "Quantum Spacetime Imprints: The 24-Cell, Standard Model Symmetry and Its Flavor Mixing" [arxiv: 2511.10685] ================================================================================ THESIS: The 24-cell acts as both the quantum of spacetime AND as a geometric representation of elementary particles. VERTEX COORDINATES USED (same as our engine): Short roots: V1 = {+/-e_i} (8 vertices) = our Form A axis-aligned Half-integer: V2 = {1/2(+/-1,+/-1,+/-1,+/-1)} (16 vertices) = our Form A half-integer Form B: permutations of (+/-1,+/-1,0,0) = our Form B HYPERCHARGE ASSIGNMENTS (derived from 24-cell geometry): Left-handed lepton doublet: Y = -1/2 Left-handed quark doublet: Y = +1/6 Right-handed charged lepton: Y = -1 Right-handed up quark: Y = +2/3 Right-handed down quark: Y = -1/3 NOTE: These MATCH the Standard Model exactly. Derived, not imposed. NEUTRINO MIXING ANGLES: Ideal (perfect tetrahedron): theta_13 = 0 deg theta_12 = 35.3 deg theta_23 = 45.0 deg With geometric distortion (eta ~ 0.02): theta_13 ~ 8.5 deg (reactor angle, matches experiment) theta_12 ~ 34 deg theta_23 ~ 45 deg Experimental values for comparison: theta_13 = 8.6 deg (Daya Bay) theta_12 = 33.4 deg theta_23 = 49 deg (some tension with prediction) KEY GEOMETRIC NUMBER: Tetrahedral angle: arccos(-1/3) = 109.47 deg NOTE: arccos(1/3) = 70.53 deg is the COMPLEMENT. This is the SAME angle we matched to Mercury's rhombohedral angle. Ali uses it for neutrino mixing. We use it for crystal structure. SAME NUMBER, DIFFERENT PHYSICAL DOMAIN. STRUCTURAL NUMBERS: 576 regular tetrahedra in the 24-cell F4 root system: 48 roots (24 short + 24 long) Weyl group W(F4): order 1152 Binary tetrahedral group T': 24 elements A4 (tetrahedral symmetry): 12 elements Spinfoam edge label: j_e = 1/2 DISTORTION PARAMETER: eta ~ 0.02-0.03 (mean deviation from perfect tetrahedron) This parameterizes how the ideal 24-cell deforms to produce realistic mixing angles. COMPARE to our C_potential alpha ~ 0.1. NO MASS PREDICTIONS — provides mixing patterns and hypercharge only. ================================================================================ 2. SINGH (2025) — "Fermion Mass Ratios from the Exceptional Jordan Algebra" [arxiv: 2508.10131, TIFR] ================================================================================ THESIS: Octonionic triality in the exceptional Jordan algebra J3(O_C) explains three fermion generations AND their mass hierarchies. UNIVERSAL PARAMETER: delta^2 = 3/8 (derived from Jordan algebra structure) This sets the eigenvalue spread (q-delta, q, q+delta) across all charged fermion sectors. FIRST-GENERATION MASS RATIOS (parameter-free prediction): sqrt(m_e) : sqrt(m_u) : sqrt(m_d) = 1 : 2 : 3 From trace split: Tr(X_l) : Tr(X_u) : Tr(X_d) = 1 : 2 : 3 CROSS-FAMILY PREDICTION: sqrt(m_tau / m_mu) = sqrt(m_s / m_d) Links lepton and quark hierarchies via E6 Dynkin automorphism. KOIDE RELATION: Pre-triality breaking: K = 2/3 (exact) Post-breaking (theoretical): K_th ~ 0.66916 Experimental: K_exp = 0.66662 (charged leptons) MATCH: Within 0.4% CKM MIXING: Cabibbo phase: phi_12 = pi/2 (geometrically fixed) Required adjustment: epsilon ~ -26.123 deg (single parameter) Cross-family normalization: kappa_23 ~ 0.55 NEUTRINO SECTOR: CP violation phase: delta_CP = +/- pi/2 (maximal, predicted) Majorana neutrinos required by the algebra. CLEBSCH-GORDAN WEIGHTS: (2, 1, 1) on the Sym^3(3) ladder These are FIXED by the algebra, not fitted. KEY NUMBERS FOR OUR COMPARISON: delta^2 = 3/8 = 0.375 Our optimal decoherence ratio: t/T ~ 0.3 (TLT-019 verified) Our collapse boundary: t/T = 0.5 = 1/2 Singh's eigenvalue spread 3/8 sits BETWEEN our optimal and collapse. Coincidence or connection? Must be investigated. ================================================================================ 3. D4 TRIALITY — STRUCTURAL NUMBERS ================================================================================ D4 root system: 24 roots (= 24-cell vertices) D4 has UNIQUE triality: outer automorphism of order 3 Three 8-dimensional representations: - Vector (8_v) - Left spinor (8_s) - Right spinor (8_c) All three have dimension 8 — this is unique to D4. Triality permutes these three representations: 8_v -> 8_s -> 8_c -> 8_v (cycle of 3) PHYSICAL INTERPRETATION (multiple authors): Three triality partners -> three fermion generations 8 dimensions each -> 8-component spinor (matches our vertex coord = 8) 24-cell decomposes into 3 tesseracts OR 3 16-cells (D4 triality) 24 / 3 = 8 (vertices per tesseract) Our observation: {3} x {8} decomposition = cipher Letters 1+2 F4 = D4 + 24 additional roots F4 has 48 roots = 24-cell + dual 24-cell The two dual 24-cells ARE Form A + Form B This is exactly our engine's source configuration vs observation target. ================================================================================ 4. F4 WEYL GROUP — KEY FACTORIZATION ================================================================================ W(F4) = 1152 = 2^7 x 3^2 Factor analysis: 2^7 = 128 3^2 = 9 128 x 9 = 1152 Subgroup structure (12 normal subgroups): Orders: 1, 2, 32, 96, 96, 192, 192, 288, 576, 576, 576, 1152 ALL subgroup orders factor into {2,3} only: 1 = 1 2 = 2 32 = 2^5 96 = 2^5 x 3 192 = 2^6 x 3 288 = 2^5 x 3^2 576 = 2^6 x 3^2 1152 = 2^7 x 3^2 NO FACTOR OF 5 appears in ANY subgroup order. Pure {2,3} at every structural level — consistent with TLT filter. ================================================================================ 5. COMPARISON TABLE — OUR NUMBERS vs PRIOR ART ================================================================================ OUR VALUE PRIOR ART VALUE MATCH? --------------------------------------------------------------------------- 24 vertices (Form A) 24 roots of D4 EXACT 24 vertices (Form B) 24 dual roots EXACT 45 deg isoclinic rotation SU(2) = unit quaternions KNOWN MATH arccos(1/3) = 70.53 deg Ali: arccos(-1/3) = 109.47 COMPLEMENT Mercury rhombohedral Ali: neutrino mixing SAME ANGLE 1152 = 2^7 x 3^2 F4 Weyl group EXACT {2,3} pure at all levels All subgroup orders EXACT 3 tesseracts decomp D4 triality (3 x 8) EXACT Vertex coord = 8 8-dim representations EXACT t/T = 0.3 optimal Singh: delta^2 = 3/8=0.375 NEARBY (25% off) t/T = 0.5 collapse Singh: q+delta ceiling STRUCTURAL? c_4D = 1.700-1.732 No prior art prediction NOVEL Fibonacci divergence No prior art observation NOVEL Geometric dominance No prior art framework NOVEL Dual channel decoherence No prior art concept NOVEL ================================================================================ 6. WHAT WE CAN USE FROM PRIOR ART ================================================================================ A. Ali's hypercharge assignments: If the 24-cell geometry derives Standard Model hypercharges, this is independent confirmation that the 24-cell IS physically meaningful. We arrived via {2,3} filtering; Ali arrives via quantum spacetime. Same geometry, same physics. B. Singh's delta^2 = 3/8: This is tantalizingly close to our decoherence optimal (0.3 vs 0.375). If the Jordan eigenvalue spread IS the decoherence ratio expressed in algebraic language, it would connect our wave-interference framework to algebraic quantum theory. C. The arccos(1/3) angle appearing in BOTH crystal structures (Mercury) and neutrino mixing (Ali) means this angle is a fundamental property of the 24-cell that manifests across scales. Same geometry, different observable consequences at different energy scales. D. D4 triality producing exactly 3 generations from 3 x 8-dim representations is independent support for our {3} x {8} cipher decomposition of the 24-cell. E. All F4 Weyl subgroup orders being pure {2,3} is independent confirmation of the {2,3} universality at the 4D level. ================================================================================