Published March 21, 2026
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Paper C v3.0: Lattice Unification Impedance, Lepton Mass Hierarchy, Mixing Angles, Cosmological Partition, and Muon Lifetime from FCC Topology
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Description
This paper consolidates all results of Ontological Resolution Theory (Canon v27.0) into a single document, deriving 22 quantities across eight sectors of particle physics and cosmology from the geometry of a single FCC lattice with coordination number k = 12. All results are parameter-free.
Eight Sectors
- Electromagnetic: Bare impedance Z_geom = 137.036 (Explicit). Precision impedance Z_phys = 137.03599919 with observer correction (Ansatz, 0.8 ppb agreement with CODATA).
- Lepton masses: Three-generation hierarchy from jet tower (Exact). Cascade operators 4φ³ and 12·2πe give mass ratios (Ansatz, 99.1–99.9%). Sieve Invariance: static and dynamic paths agree within aperture gap Δ = 2πe − 4φ³ ≈ 0.135 (Explicit).
- Electroweak: VEV = m_τ · Z_geom · (1 + 1/89) = 246.23 GeV (Ansatz, 99.996%). Fermi coupling G_F = 1/(√2 · VEV²) = 1.165 × 10⁻⁵ GeV⁻² (Explicit, 99.9%). Weinberg angle sin²θ_W = 3/13 = 0.2308 (Ansatz, 99.8%).
- Flavour — PMNS: Solar angle sin²θ₁₂ = 4/13 = 0.3077 (Ansatz, centre of 1σ). Reactor angle sin²θ₁₃ = 3/132 = 1/44 = 0.02273 (Ansatz, boundary of 1σ).
- Flavour — CKM: Cabibbo angle sinθ_C = 2/9 + 1/288 = 0.22569 (Ansatz, centre of 1σ).
- Strong sector: α_s(M_Z) = 16/Z_geom = 0.1168 (Explicit, 99%). N_c = 3 (Explicit). N_gluon = k − 4 = 8 (Explicit).
- Cosmology: Topological denominator W_total = 2(k + R_max) = 42 (Explicit). Baryon fraction Ω_b = 2/42 = 1/21 (Explicit, 97.1%). Dark matter Ω_DM = 11/42 (Explicit, 97.7%). Dark energy Ω_Λ = 29/42 (Explicit, 99.0%). Ratio Ω_DM/Ω_b = 11/2 = 5.5 (Explicit, 99.5%).
- Dynamics: Time dilation dτ/dt = 1 − W/Z_geom (Explicit). Electron absolutely stable (Explicit). Phase-space identity 192π³ = (4π)³·3 (Explicit). Muon lifetime τ_μ = 2.27 × 10⁻⁶ s (Ansatz, 96.8%). Lifetime ratio τ_μ/τ_τ = 7.43 × 10⁶ (Ansatz, 97.8%).
Key Results Table
| Quantity | Formula | ORT | Experiment | Status |
|---|---|---|---|---|
| Z_phys | Z_bare − Δ₁ + Δ₁/(k−3) | 137.03599919 | 137.03599908 | Ansatz (0.8 ppb) |
| m_τ/m_μ | 4φ³ | 16.944 | 16.817 | Ansatz (99.25%) |
| m_μ/m_e | 12·2πe | 204.95 | 206.768 | Ansatz (99.12%) |
| VEV | m_τ · Z_geom · (90/89) | 246.23 GeV | 246.22 GeV | Ansatz (99.996%) |
| G_F | 1/(√2 · VEV²) | 1.165 × 10⁻⁵ | 1.166 × 10⁻⁵ | Explicit (99.9%) |
| sin²θ_W | 3/13 | 0.2308 | 0.2312 | Ansatz (99.8%) |
| sin²θ₁₂ | 4/13 | 0.3077 | 0.307 ± 0.013 | Ansatz (1σ) |
| sin²θ₁₃ | 3/132 | 0.02273 | 0.0220 ± 0.0007 | Ansatz (1σ) |
| sinθ_C | 2/9 + 1/288 | 0.22569 | 0.2257 ± 0.0006 | Ansatz (1σ) |
| α_s(M_Z) | 16/Z_geom | 0.1168 | 0.1179 | Explicit (99%) |
| Ω_b | 2/42 | 0.0476 | 0.0490 | Explicit (97.1%) |
| Ω_DM | 11/42 | 0.2619 | 0.268 | Explicit (97.7%) |
| Ω_Λ | 29/42 | 0.6905 | 0.683 | Explicit (99.0%) |
| Ω_DM/Ω_b | 11/2 | 5.50 | 5.47 | Explicit (99.5%) |
| τ_μ | Master formula | 2.27 μs | 2.20 μs | Ansatz (96.8%) |
| τ_μ/τ_τ | (k−1)/2 · P_τ/P_μ · (w_τ/w_μ)⁴ | 7.43 × 10⁶ | 7.6 × 10⁶ | Ansatz (97.8%) |
| 192π³ | (4π)^d · d | Identity | 192π³ | Explicit |
| Electron stability | Locality + charge | p_D = 0 | > 10²⁸ yr | Explicit |
| dτ/dt | 1 − W/Z_geom | Derived | GR limit | Explicit |
Universal Numbers from k = 12
| Number | Definition | Sectors |
|---|---|---|
| 13 = k+1 | Effective coordination | Weinberg angle (3/13), solar neutrino (4/13) |
| 11 = k−1 | Vacancy invariant | Impedance (Δ₁), dark matter (11/42), reactor angle (3/132), evacuation channels (11/2) |
| 9 = k−3 | Charged modes | Impedance relaxation (Δ₂), Cabibbo angle (2/9), R_max, strong sector |
| 3 | Neutral axes / d | Weinberg (3/13), reactor (3/132), N_c = 3, phase-space (4π)³·3 |
| 42 = 2(k+R_max) | Topological denominator | Cosmological partition |
| 89 = F₁₁ | Fibonacci predecessor | VEV uplift (1/89) |
| 144 = k² = F₁₂ | Coordination squared | Vertex impedance (1/144), Fibonacci lock |
What Changed from v1.0 to v2.0
| Aspect | v1.0 (Canon v15.0) | v2.0 (Canon v27.0) |
|---|---|---|
| Canon version | v15.0 | v27.0 |
| m_τ/m_μ | 2πe(1 − 2/k²) = 16.848 | 4φ³ = 16.944 |
| m_μ/m_e | 12·2πe(1 + 1/k²) = 206.45 | 12·2πe = 204.95 |
| m_τ/m_e | (π³−6)Z + 4π² + 11(1+1/4π²) = 3477.228 | Sieve Invariance: two paths (3440 / 3473) |
| α_s | √2 · Z_1D(1+13/Z) = 0.1190 | 16/Z_geom = 0.1168 |
| VEV | Not derived | m_τ · Z_geom · (1+1/89) = 246.23 GeV (NEW) |
| G_F | Not derived | 1/(√2 · VEV²) = 1.165 × 10⁻⁵ GeV⁻² (NEW) |
| Cosmology | Not included | Full partition: 2/42, 11/42, 29/42 (NEW) |
| Muon lifetime | Not included | 2.27 μs from master formula (NEW) |
| Time dilation | Not included | dτ/dt = 1 − W/Z_geom (NEW) |
| Electron stability | Not included | Proved from locality + charge (NEW) |
| Phase-space identity | Not included | 192π³ = (4π)³·3 (NEW) |
| Pair Law | Axiom (postulated) | Derived from Z₂ symmetry (Exact) |
| Statuses | Mixed (Theorem for Ansatz) | Honest: Exact/Explicit/Ansatz per rules |
| Total quantities | 14 | 22 |
| Sectors covered | 6 | 8 |
| Mixing angles | θ₁₂, θ₁₃, θ_C (retained) | Same formulas, updated status to Ansatz |
| Precision impedance | 0.8 ppb (retained) | Same formula, updated status to Ansatz |
Summary of Changes
- 8 new results added: VEV, G_F, cosmological partition (3 fractions), muon lifetime, time dilation, electron stability, phase-space identity
- All mass formulas updated from Canon v15.0 operators to Canon v27.0 operators (4φ³ replaces 2πe corrections; 16/Z_geom replaces √2·Z_1D formula for α_s)
- Pair Law upgraded from postulate to Source-level derivation via Z₂ symmetry
- All statuses corrected per ORT epistemic rules: Ansatz is never called Theorem
- Sieve Invariance replaces single-formula tau mass with dual-path geometric invariant
- Bibliography updated to Canon v27.0, all Papers referenced with DOIs where available
- Total result count: 14 → 22. Sector count: 6 → 8
Falsification Criteria
- Z_phys: deviation > 10⁻⁷ from formula (precision QED, g−2)
- sin²θ₁₂: value outside 0.29–0.32 (JUNO)
- sin²θ₁₃: value outside 0.020–0.025 (JUNO, Hyper-K)
- sinθ_C: value outside 0.224–0.227 (LHCb, BESIII)
- Ω_b: value outside 0.044–0.052 (CMB Stage-4)
- τ_μ: value outside 2.1–2.4 μs (MuLan)
- Lorentz violation: none at E > 10²⁰ GeV (Fermi LAT, CTA)
Open Problems
- M-1: VEV from pure geometry (Ansatz at 99.996%, pending formal derivation of 1/F₁₁)
- L1: Absolute neutrino masses
- L2: Atmospheric angle θ₂₃ and CP phase δ_CP
- L4: Exact α without Kepler bridge
- L5: Running α_s(Q²)
- L7: Small CKM angles θ₂₃, θ₁₃
- N-1: Lorentz violation parameter η
- N-2: Explicit rewrite rule R
- C-1: Formal derivation of Ω_b = 1/(k + R_max)
- C-2: Independent derivation of 29 = 24 + 5
Dependencies
Paper Zero v1.1 · Paper A v9.1 · Paper B v2.0 · Paper ED v1.0 · Paper G v1.2 · Paper M v3.0 · Paper N v2.0 · Paper Q v2.1 · Paper S v2.0 · Paper U v1.0 · Dark Matter Letter v1.0 · Canon v27.0
One lattice. One integer k = 12. Twenty-two quantities. Zero free parameters. Eight sectors of physics. The lattice speaks.