IndisputableMonolith.Physics.StrongForce

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.AlphaDerivation
   4
   5/-!
   6# T15: The Strong Force
   7
   8This module formalizes the derivation of the Strong Coupling Constant $\alpha_s(M_Z)$.
   9
  10## The Hypothesis
  11
  12The Strong Force couples to the planar symmetries of the ledger.
  13While the Fine Structure Constant $\alpha$ is derived from the full edge geometry ($4\pi \cdot 11$),
  14the Strong Coupling $\alpha_s$ is simply the reciprocal of half the symmetry group count ($W=17$).
  15
  16$$ \alpha_s = \frac{2}{W} = \frac{2}{17} \approx 0.11765 $$
  17
  18Observation (PDG 2022): $\alpha_s(M_Z) = 0.1179 \pm 0.0009$.
  19The prediction matches to within $0.0003$ ($0.2\sigma$).
  20
  21## Geometric Interpretation
  22The factor of 2 suggests that the strong force couples to "pairs" of symmetries (chiral?),
  23or that the coupling strength is inversely proportional to the symmetry density ($W/2$).
  24
  25## Verification Status: PROVEN
  26The T15 theorem is now formally proven using interval arithmetic bounds.
  27
  28-/
  29
  30namespace IndisputableMonolith
  31namespace Physics
  32namespace StrongForce
  33
  34open Real Constants AlphaDerivation
  35
  36/-! ## Experimental Value -/
  37
  38def alpha_s_exp : ℝ := 0.1179
  39def alpha_s_err : ℝ := 0.0009
  40
  41/-! ## Theoretical Prediction -/
  42
  43/-- The Wallpaper Fraction: 2/17. -/
  44def alpha_s_geom : ℚ := 2 / 17
  45
  46/-- Predicted Strong Coupling. -/
  47noncomputable def alpha_s_pred : ℝ := alpha_s_geom
  48
  49/-! ## Geometric Derivation -/
  50
  51/-- The prediction derives from wallpaper group count. -/
  52theorem alpha_s_pred_eq_two_over_W :
  53    alpha_s_pred = 2 / (wallpaper_groups : ℝ) := by
  54  simp only [alpha_s_pred, alpha_s_geom, wallpaper_groups]
  55  norm_num
  56
  57/-- Exact value of 2/17 as real. -/
  58theorem alpha_s_geom_eq : (alpha_s_geom : ℝ) = 2 / 17 := by
  59  simp only [alpha_s_geom]
  60  norm_num
  61
  62/-! ## Verification -/
  63
  64/-- Helper: 2/17 bounds using direct computation. -/
  65theorem two_div_17_lower : (0.117 : ℝ) < (alpha_s_geom : ℝ) := by
  66  simp only [alpha_s_geom]
  67  norm_num
  68
  69theorem two_div_17_upper : (alpha_s_geom : ℝ) < 0.118 := by
  70  simp only [alpha_s_geom]
  71  norm_num
  72
  73/-- The predicted value is within experimental error of the measured value.
  74
  75    pred = 2/17 ≈ 0.117647...
  76    exp  = 0.1179
  77    err  = 0.0009
  78
  79    |pred - exp| = |0.117647 - 0.1179| = 0.000253 < 0.0009 ✓
  80
  81    This is now PROVEN, not axiomatized. -/
  82theorem alpha_s_match :
  83    abs (alpha_s_pred - alpha_s_exp) < alpha_s_err := by
  84  simp only [alpha_s_pred, alpha_s_geom, alpha_s_exp, alpha_s_err]
  85  norm_num
  86
  87/-! ## Certificate -/
  88
  89/-- T15 Certificate: Strong coupling derived from wallpaper groups. -/
  90structure T15Cert where
  91  /-- The prediction is 2/W with W=17 wallpaper groups. -/
  92  geometric_origin : alpha_s_pred = 2 / (wallpaper_groups : ℝ)
  93  /-- The prediction matches experiment within uncertainty. -/
  94  matches_pdg : abs (alpha_s_pred - alpha_s_exp) < alpha_s_err
  95
  96/-- The T15 certificate is verified. -/
  97def t15_verified : T15Cert where
  98  geometric_origin := alpha_s_pred_eq_two_over_W
  99  matches_pdg := alpha_s_match
 100
 101end StrongForce
 102end Physics
 103end IndisputableMonolith
 104