IndisputableMonolith.Constants.FineStructureConstant

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Foundation.PhiForcing
   4
   5/-!
   6# C-001: Fine Structure Constant Derivation
   7
   8Formalizes the RS derivation of the fine structure constant α from φ.
   9
  10## Registry Item
  11- C-001: What determines the fine structure constant α ≈ 1/137?
  12
  13## RS Derivation
  14α_lock = (1 − 1/φ)/2. This is the canonical coupling in RS-native units,
  15derived from the ledger's reciprocal symmetry J(x) = J(1/x).
  16
  17The conventional α ≈ 1/137.036 relates to α_lock via the full
  18ledger-to-lab conversion (see Physics/AlphaPrecision.lean).
  19-/
  20
  21namespace IndisputableMonolith
  22namespace Constants
  23namespace FineStructureConstant
  24
  25open Real Constants
  26
  27/-! ## Definition and Basic Facts -/
  28
  29/-- α_lock > 0 (re-export from Constants). -/
  30theorem alphaLock_pos : 0 < alphaLock := Constants.alphaLock_pos
  31
  32/-- α_lock < 1 (re-export from Constants). -/
  33theorem alphaLock_lt_one : alphaLock < 1 := Constants.alphaLock_lt_one
  34
  35/-- α_lock lies in the open unit interval. -/
  36theorem alphaLock_in_unit_interval : 0 < alphaLock ∧ alphaLock < 1 :=
  37  ⟨alphaLock_pos, alphaLock_lt_one⟩
  38
  39/-! ## Numerical Bounds -/
  40
  41/-- α_lock is between 0.18 and 0.21 (coarse bound from φ ∈ (1.61, 1.62)). -/
  42theorem alphaLock_numerical_bounds :
  43    (0.18 : ℝ) < alphaLock ∧ alphaLock < (0.21 : ℝ) := by
  44  unfold alphaLock
  45  have h_phi := phi_gt_onePointSixOne
  46  have h_phi' := phi_lt_onePointSixTwo
  47  constructor
  48  · have h_inv : 1 / phi < 1 / 1.61 := by
  49      rw [div_lt_div_iff_of_pos_left (by norm_num) phi_pos (by norm_num)]
  50      exact h_phi
  51    linarith
  52  · have h_inv : 1 / 1.62 < 1 / phi := by
  53      rw [div_lt_div_iff_of_pos_left (by norm_num) (by norm_num) phi_pos]
  54      exact h_phi'
  55    linarith
  56
  57/-! ## C-001 Resolution Statement -/
  58
  59/-- **C-001 Resolution**: The fine structure constant is determined by φ.
  60
  61    α_lock = (1 − 1/φ)/2 has no free parameters.
  62    It is the unique coupling compatible with the ledger's
  63    reciprocal symmetry and self-similar closure.
  64
  65    Relationship to measured α ≈ 1/137 requires the conversion
  66    from RS-native to SI units (λ_rec calibration). -/
  67theorem fine_structure_derived :
  68    0 < alphaLock ∧ alphaLock < 1 ∧
  69    alphaLock = (1 - 1 / phi) / 2 :=
  70  ⟨alphaLock_pos, alphaLock_lt_one, rfl⟩
  71
  72end FineStructureConstant
  73end Constants
  74end IndisputableMonolith
  75