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GAUGE INVARIANCE TEST SUITE - SUMMARY
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File: simulations/tests/test_gauge_invariance.py
Lines of Code: 820
Total Tests: 18
Test Classes: 5
All Tests: PASSING

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TEST COVERAGE
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1. TestGaugeGroupStructure (4 tests)
   ✓ test_gauge_coupling_unification_at_gut_scale
     → Verifies α_GUT ≈ 1/24 from G2 topology (b₃=24)
     → Validates M_GUT ≈ 2×10^16 GeV

   ✓ test_gauge_group_dimension_consistency
     → SM: dim = 12 = 8(SU3) + 3(SU2) + 1(U1)
     → SU(5): dim = 24
     → SO(10): dim = 45

   ✓ test_g2_holonomy_to_su3_embedding
     → G2 (dim=14, rank=2) ⊃ SU(3) (dim=8, rank=2)
     → TCS G2 manifold: b₃ = 24

   ✓ test_so10_fermion_representation
     → 16-spinor = one complete generation
     → 3 generations = 48 fermions (including ν_R)

2. TestSymmetryBreakingChain (4 tests)
   ✓ test_gut_symmetry_breaking_scale
     → SO(10) → SU(3)×SU(2)×U(1) at M_GUT
     → Proton lifetime: τ_p > 1.6×10^34 years

   ✓ test_electroweak_symmetry_breaking
     → SU(2)×U(1) → U(1)_EM at v_EW = 246 GeV
     → W and Z boson masses consistent
     → m_Z = m_W / cos(θ_W)

   ✓ test_quantum_number_conservation_at_breaking
     → 16_SO(10) → (3,2,1/6) + ... (6 representations)
     → Q = T³ + Y conserved throughout

   ✓ test_weak_mixing_angle_running
     → sin²θ_W: 3/8 (GUT) → 0.2312 (M_Z)
     → Validates RG running direction

3. TestAnomalyCancellation (4 tests)
   ✓ test_triangle_anomaly_cancellation_U1
     → U(1)³ anomaly: Σ n_f Y_f³ = 0
     → Quarks + leptons cancel to <10^-10

   ✓ test_triangle_anomaly_cancellation_SU2_U1
     → SU(2)²×U(1) anomaly: Σ T(R_f) Y_f = 0
     → Only doublets contribute

   ✓ test_gravitational_anomaly_cancellation
     → n_left - n_right = 0
     → Requires right-handed neutrinos (SO(10) feature)

   ✓ test_so10_automatic_anomaly_cancellation
     → All anomalies cancel within 16-spinor
     → No fine-tuning needed

4. TestHyperchargeQuantization (3 tests)
   ✓ test_hypercharge_quantization_condition
     → All Y ∈ ℚ (rational numbers)
     → Y ∈ {0, ±1/6, ±1/3, ±1/2, ±2/3, 1}

   ✓ test_hypercharge_and_charge_consistency
     → Q = T³ + Y (Gell-Mann-Nishijima)
     → Quark charges: {-1/3, 2/3}
     → Lepton charges: {-1, 0}

   ✓ test_hypercharge_normalization_in_gut
     → GUT normalization: Y_GUT = √(3/5) × Y_SM
     → Factor (5/3) for SU(5) embedding

5. TestGaugeInvarianceUnderRGEvolution (3 tests)
   ✓ test_beta_function_coefficients
     → b₁ = 41/10 (U(1)_Y)
     → b₂ = -19/6 (SU(2)_L)
     → b₃ = -7 (SU(3)_c - asymptotic freedom)

   ✓ test_asymptotic_freedom_su3
     → b₃ = 11 - 2n_f/3 = 7 (n_f = 6 flavors)
     → α₃ decreases at high energy

   ✓ test_gauge_coupling_threshold_corrections
     → KK tower at M_* ≈ 5 TeV
     → Δ(1/α_i) = (k_i h^{1,1})/(2π) log(M_GUT/M_*)
     → h^{1,1} = 24 from TCS G2 topology

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KEY PHYSICS VALIDATIONS
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✓ G2 Holonomy → SO(10) → Standard Model
  - Gauge group emerges naturally from G2 structure
  - b₃ = 24 determines α_GUT ≈ 1/24

✓ Grand Unification at M_GUT ≈ 2×10^16 GeV
  - From G2 torsion and moduli stabilization
  - Consistent with proton decay bounds

✓ Anomaly-Free Theory
  - All triangle anomalies cancel
  - Gravitational anomaly vanishes
  - SO(10) provides automatic cancellation

✓ Right-Handed Neutrinos Required
  - Needed for gravitational anomaly cancellation
  - Naturally included in SO(10) 16-spinor
  - Explains neutrino masses via seesaw mechanism

✓ Threshold Corrections from KK Tower
  - h^{1,1} = 24 Hodge number from G2 manifold
  - Corrections modify gauge running above M_* ≈ 5 TeV
  - Consistent with dimensional reduction

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DEPENDENCIES
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Required packages:
  - pytest >= 8.4.1
  - numpy >= 1.24.0

PM framework modules:
  - simulations.base.PMRegistry
  - simulations.base.EstablishedPhysics

Data sources:
  - PDG 2024 (gauge couplings, particle masses)
  - NuFIT 6.0 (neutrino mixing parameters)
  - Super-Kamiokande (proton decay bounds)

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USAGE
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Run all tests:
  $ pytest simulations/tests/test_gauge_invariance.py -v

Run specific test class:
  $ pytest simulations/tests/test_gauge_invariance.py::TestAnomalyCancellation -v

Run with gauge marker:
  $ pytest -m gauge -v

Run as standalone script:
  $ python simulations/tests/test_gauge_invariance.py

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REFERENCES
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[1] Langacker (1981): "Grand Unified Theories and Proton Decay"
    Phys. Rep. 72, 185-385

[2] PDG 2024: Particle Data Group
    https://pdg.lbl.gov

[3] Joyce (2007): "Riemannian Holonomy Groups and Calibrated Geometry"
    Oxford Mathematical Monographs

[4] TCS G2 Manifold: Third Betti number b₃ = 24 (manifold #187)

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Author: Andrew Keith Watts
Copyright (c) 2025-2026. All rights reserved.
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