IndisputableMonolith.Physics.CKMGeometry

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.PhiSupport
   4import IndisputableMonolith.Constants.AlphaDerivation
   5import IndisputableMonolith.Constants.Alpha
   6import IndisputableMonolith.Numerics.Interval.PhiBounds
   7import IndisputableMonolith.Numerics.Interval.AlphaBounds
   8import IndisputableMonolith.Physics.MixingGeometry
   9
  10/-!
  11# T11: The CKM Matrix Geometry
  12
  13This module formalizes the derivation of the CKM mixing angles from the
  14ledger geometry and the fine-structure constant.
  15
  16## The Hypothesis
  17
  18The CKM matrix elements $|V_{us}|, |V_{cb}|, |V_{ub}|$ are not arbitrary parameters.
  19They are determined by geometric couplings on the cubic ledger:
  20
  211. **Theta 13 ($V_{ub}$)**: The "Fine Structure Coupling".
  22   $$ |V_{ub}| = \frac{\alpha}{2} $$
  23   Prediction: 0.00365. Observation: 0.00369(11). Match: < 1 sigma.
  24
  252. **Theta 23 ($V_{cb}$)**: The "Edge-Dual Coupling".
  26   $$ |V_{cb}| = \frac{1}{2 E_{total}} = \frac{1}{24} $$
  27   Prediction: 0.04167. Observation: 0.04182(85). Match: < 0.2 sigma.
  28
  293. **Theta 12 ($V_{us}$, Cabibbo)**: The "Golden Projection".
  30   $$ |V_{us}| = \phi^{-3} - \frac{3}{2}\alpha $$
  31   Prediction: 0.2251. Observation: 0.2250(7). Match: < 0.2 sigma.
  32
  33## Verification Status
  34
  35T11 V_cb theorem is now PROVEN (1/24 is exact rational).
  36V_ub and V_us depend on α and φ which require interval arithmetic for full proofs.
  37
  38-/
  39
  40namespace IndisputableMonolith
  41namespace Physics
  42namespace CKMGeometry
  43
  44open Real Constants AlphaDerivation MixingGeometry
  45
  46/-! ## Experimental Values (PDG 2022) -/
  47
  48def V_us_exp : ℝ := 0.22500
  49def V_us_err : ℝ := 0.00067
  50
  51def V_cb_exp : ℝ := 0.04182
  52def V_cb_err : ℝ := 0.00085
  53
  54def V_ub_exp : ℝ := 0.00369
  55def V_ub_err : ℝ := 0.00011
  56
  57/-! ## Theoretical Predictions -/
  58
  59/-- V_ub is half the fine-structure constant (fine_structure_leakage). -/
  60noncomputable def V_ub_pred : ℝ := fine_structure_leakage
  61
  62/-- V_cb geometric ratio: 1/vertex_edge_slots = 1/24. -/
  63def V_cb_geom : ℚ := edge_dual_ratio
  64
  65/-- V_cb is the inverse of twice the total edge count (1/24). -/
  66noncomputable def V_cb_pred : ℝ := (V_cb_geom : ℝ)
  67
  68/-- V_us is the golden projection (torsion_overlap) minus a radiative correction. -/
  69noncomputable def V_us_pred : ℝ := torsion_overlap - cabibbo_radiative_correction
  70
  71/-! ## Geometric Derivation -/
  72
  73/-- V_cb derives from cube edge geometry: 1/(2 * 12) = 1/24. -/
  74theorem V_cb_from_cube_edges :
  75    V_cb_geom = 1 / (2 * cube_edges 3) := by
  76  simp only [V_cb_geom, edge_dual_ratio, cube_edges]
  77  norm_num
  78
  79/-! ## Verification Theorems -/
  80
  81/-- V_cb matches within 1 sigma.
  82
  83    pred = 1/24 ≈ 0.04166666...
  84    obs  = 0.04182
  85    err  = 0.00085
  86    |pred - obs| = |0.04166 - 0.04182| = 0.00016 < 0.00085 ✓
  87
  88    This is now PROVEN, not axiomatized. -/
  89theorem V_cb_match : abs (V_cb_pred - V_cb_exp) < V_cb_err := by
  90  simp only [V_cb_pred, V_cb_geom, V_cb_exp, V_cb_err, edge_dual_ratio]
  91  norm_num
  92
  93/-- Bounds on alpha needed for CKM proofs.
  94    alphaInv ≈ 137.036 so alpha ≈ 0.00730
  95    NOTE: These bounds are verified numerically but require transcendental
  96    computation (involving π and ln(φ)) that norm_num cannot handle. -/
  97theorem alpha_lower_bound : (0.00729 : ℝ) < Constants.alpha := by
  98  -- From the rigorous interval proof: alphaInv < 137.039 ⇒ 1/137.039 < alpha
  99  have h_inv_lt : Constants.alphaInv < (137.039 : ℝ) := by
 100    simpa [Constants.alphaInv] using (IndisputableMonolith.Numerics.alphaInv_lt)
 101  have h_inv_pos : (0 : ℝ) < Constants.alphaInv := by
 102    have h := IndisputableMonolith.Numerics.alphaInv_gt
 103    linarith
 104  -- Invert inequality (antitone on positive reals).
 105  have h_one_div : (1 / (137.039 : ℝ)) < 1 / Constants.alphaInv := by
 106    exact one_div_lt_one_div_of_lt h_inv_pos h_inv_lt
 107  -- Translate to alpha = 1/alphaInv and weaken the numeric constant to 0.00729.
 108  have h_num : (0.00729 : ℝ) < (1 / (137.039 : ℝ)) := by norm_num
 109  simpa [Constants.alpha, one_div] using lt_trans h_num h_one_div
 110
 111theorem alpha_upper_bound : Constants.alpha < (0.00731 : ℝ) := by
 112  -- From the rigorous interval proof: 137.030 < alphaInv ⇒ alpha < 1/137.030
 113  have h_inv_gt : (137.030 : ℝ) < Constants.alphaInv := by
 114    simpa [Constants.alphaInv] using (IndisputableMonolith.Numerics.alphaInv_gt)
 115  have h_pos : (0 : ℝ) < (137.030 : ℝ) := by norm_num
 116  -- Invert inequality (antitone on positive reals): 1/alphaInv < 1/137.030
 117  have h_one_div : (1 / Constants.alphaInv) < 1 / (137.030 : ℝ) := by
 118    exact one_div_lt_one_div_of_lt h_pos h_inv_gt
 119  -- Translate to alpha = 1/alphaInv and weaken the numeric constant to 0.00731.
 120  have h_num : (1 / (137.030 : ℝ)) < (0.00731 : ℝ) := by norm_num
 121  have : Constants.alpha < 1 / (137.030 : ℝ) := by
 122    simpa [Constants.alpha, one_div] using h_one_div
 123  exact lt_trans this h_num
 124
 125/-- V_ub matches within 1 sigma.
 126
 127    V_ub_pred = alpha/2 ≈ 0.00365
 128    V_ub_exp = 0.00369
 129    |V_ub_pred - V_ub_exp| ≈ 0.00004 < 0.00011 ✓
 130
 131    Proof: From alpha bounds (0.00729, 0.00731), we get
 132    alpha/2 ∈ (0.003645, 0.003655), and
 133    |0.00365 - 0.00369| = 0.00004 < 0.00011. -/
 134theorem V_ub_match : abs (V_ub_pred - V_ub_exp) < V_ub_err := by
 135  have h_alpha_lower := alpha_lower_bound
 136  have h_alpha_upper := alpha_upper_bound
 137  simp only [V_ub_pred, V_ub_exp, V_ub_err, fine_structure_leakage]
 138  -- alpha/2 ∈ (0.003645, 0.003655)
 139  have h_lower : (0.003645 : ℝ) < Constants.alpha / 2 := by linarith
 140  have h_upper : Constants.alpha / 2 < (0.003655 : ℝ) := by linarith
 141  -- |alpha/2 - 0.00369| ≤ max(|0.003645 - 0.00369|, |0.003655 - 0.00369|)
 142  --                     = max(0.000045, 0.000035) = 0.000045 < 0.00011
 143  rw [abs_lt]
 144  constructor <;> linarith
 145
 146/-- Bounds on phi^(-3) needed for V_us proof.
 147    φ^(-3) ≈ 0.2360679...
 148
 149    These bounds are derived from phi_tight_bounds via antitonicity. -/
 150theorem phi_inv3_lower_bound : (0.2360 : ℝ) < phi ^ (-3 : ℤ) :=
 151  Numerics.phi_inv3_zpow_bounds.1
 152
 153theorem phi_inv3_upper_bound : phi ^ (-3 : ℤ) < (0.2361 : ℝ) :=
 154  Numerics.phi_inv3_zpow_bounds.2
 155
 156/-- V_us matches within 1 sigma.
 157
 158    V_us_pred = φ^(-3) - (3/2)*α
 159              ≈ 0.2360679 - 0.01095
 160              ≈ 0.2251
 161    V_us_exp = 0.22500
 162    |V_us_pred - V_us_exp| ≈ 0.0001 < 0.00067 ✓
 163
 164    Proof: From bounds on φ^(-3) and α, we establish the match. -/
 165theorem V_us_match : abs (V_us_pred - V_us_exp) < V_us_err := by
 166  have h_alpha_lower := alpha_lower_bound
 167  have h_alpha_upper := alpha_upper_bound
 168  have h_phi3_lower := phi_inv3_lower_bound
 169  have h_phi3_upper := phi_inv3_upper_bound
 170  simp only [V_us_pred, V_us_exp, V_us_err, torsion_overlap, cabibbo_radiative_correction]
 171  -- V_us_pred = phi^(-3) - 1.5*alpha
 172  -- phi^(-3) ∈ (0.2360, 0.2361)
 173  -- 1.5*alpha ∈ (0.010935, 0.010965)
 174  -- V_us_pred ∈ (0.2360 - 0.010965, 0.2361 - 0.010935) = (0.225035, 0.225165)
 175  have h_correction_lower : (0.010935 : ℝ) < (3/2) * Constants.alpha := by linarith
 176  have h_correction_upper : (3/2) * Constants.alpha < (0.010965 : ℝ) := by linarith
 177  have h_pred_lower : (0.225035 : ℝ) < Constants.phi ^ (-3 : ℤ) - (3/2) * Constants.alpha := by linarith
 178  have h_pred_upper : Constants.phi ^ (-3 : ℤ) - (3/2) * Constants.alpha < (0.225165 : ℝ) := by linarith
 179  -- |V_us_pred - 0.22500| ≤ max(|0.225035 - 0.22500|, |0.225165 - 0.22500|)
 180  --                      = max(0.000035, 0.000165) = 0.000165 < 0.00067
 181  rw [abs_lt]
 182  constructor <;> linarith
 183
 184/-! ## Certificate -/
 185
 186/-- T11 Certificate: V_cb from cube edge geometry. -/
 187structure T11Cert where
 188  /-- V_cb = 1/(2*12) = 1/24 from cube edges. -/
 189  geometric_origin : V_cb_geom = 1 / (2 * cube_edges 3)
 190  /-- The prediction matches experiment within uncertainty. -/
 191  matches_pdg : abs (V_cb_pred - V_cb_exp) < V_cb_err
 192
 193/-- The T11 certificate (for V_cb) is verified. -/
 194def t11_V_cb_verified : T11Cert where
 195  geometric_origin := V_cb_from_cube_edges
 196  matches_pdg := V_cb_match
 197
 198end CKMGeometry
 199end Physics
 200end IndisputableMonolith
 201