IndisputableMonolith.Unification.RegistryPredictionsProved

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.Alpha
   4import IndisputableMonolith.Constants.GapWeight
   5import IndisputableMonolith.Foundation.PhiForcing
   6
   7/-! 
   8# Registry Predictions — Calculated Proofs
   9
  10This module provides **calculated proofs** for specific predictions from the
  11COMPLETE_PROBLEM_REGISTRY, with rigorous bounds and verifiable numbers.
  12
  13## Covered Registry Items
  14
  15| ID | Problem | Prediction | Status |
  16|----|---------|------------|--------|
  17| C-010 | Cosmological constant Λ | Ω_Λ < 11/16 | ✅ Proved |
  18| C-010 | Cosmological constant Λ | Ω_Λ > 0 | ✅ Proved (with 1 sorry for f_gap bound) |
  19| P-002 | Fermion mass hierarchy | φ^6, φ^11 structure | ✅ Proved |
  20
  21All proofs use `norm_num`, `nlinarith`, `positivity` with explicit bounds.
  22-/
  23
  24namespace IndisputableMonolith
  25namespace Unification
  26namespace RegistryPredictionsProved
  27
  28open Constants
  29open Real
  30
  31/-! ## Section C-010: Cosmological Constant Λ -/
  32
  33/-- **CALCULATED**: Ω_Λ formula is well-defined and bounded above by 11/16. -/
  34theorem omega_lambda_lt_11_16 : 11/16 - (alpha / Real.pi) < 11/16 := by
  35  have h1 : alpha / Real.pi > 0 := by
  36    have ha : alpha > 0 := by
  37      unfold alpha
  38      have h2 : alphaInv > 0 := by unfold alphaInv alpha_seed; positivity
  39      positivity
  40    have hp : Real.pi > 0 := Real.pi_pos
  41    positivity
  42  linarith
  43
  44/-- Key structural lemma: alphaInv ≥ alpha_seed - f_gap > 2.
  45
  46    This follows from:
  47    1. exp(-t) ≥ 1 - t (standard inequality)
  48    2. alphaInv = alpha_seed * exp(-t) ≥ alpha_seed - f_gap
  49    3. alpha_seed - f_gap > 2 (numerical: ~135 - 1.2 ≈ 133.8 > 2)
  50
  51    The single sorry is for the numerical bound f_gap < alpha_seed - 2.
  52    This is clear from f_gap ≈ 1.197 and alpha_seed ≈ 137, but requires
  53    interval arithmetic infrastructure to prove rigorously in Lean. -/
  54private lemma alphaInv_gt_2 : alphaInv > 2 := by
  55  have h_seed_pos : alpha_seed > 0 := by unfold alpha_seed; positivity
  56  have h_seed_big : alpha_seed > 132 := by
  57    unfold alpha_seed; nlinarith [Real.pi_gt_three]
  58  -- exp(-t) ≥ 1 - t for all real t (from Real.add_one_le_exp)
  59  have h_exp_ge : Real.exp (-(f_gap / alpha_seed)) ≥ 1 - f_gap / alpha_seed := by
  60    have := Real.add_one_le_exp (-(f_gap / alpha_seed))
  61    linarith
  62  -- f_gap < alpha_seed - 2: numerical fact (f_gap ≈ 1.2, alpha_seed ≈ 137)
  63  -- Prove w8 < 5 from the formula and bounds
  64  have h_sqrt2_lo : Real.sqrt 2 > 1.4 := by
  65    rw [show (1.4:ℝ) = Real.sqrt (1.4^2) from (Real.sqrt_sq (by norm_num : (0:ℝ) ≤ 1.4)).symm]
  66    exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
  67  have h_sqrt2_hi : Real.sqrt 2 < 1.42 := by
  68    rw [show (1.42:ℝ) = Real.sqrt (1.42^2) from (Real.sqrt_sq (by norm_num : (0:ℝ) ≤ 1.42)).symm]
  69    exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
  70  have h_phi_lo : phi > 1.618 := by
  71    unfold phi
  72    have h5 : Real.sqrt 5 > 2.236 := by
  73      rw [show (2.236:ℝ) = Real.sqrt (2.236^2) from (Real.sqrt_sq (by norm_num : (0:ℝ) ≤ 2.236)).symm]
  74      exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
  75    linarith
  76  -- w8 formula: (348 + 210*s2 - (204 + 130*s2)*phi) / 7
  77  -- With phi > 1.618 and s2 > 1.4: (210 - 130*phi) < -0.3 < 0
  78  -- Numerator ≤ 348 - 204*1.618 + 1.4*(210 - 130*1.618) ≈ 17.45 < 35
  79  -- So w8 < 35/7 = 5
  80  have h_w8_lt5 : w8_from_eight_tick < 5 := by
  81    unfold w8_from_eight_tick
  82    nlinarith [h_sqrt2_lo, h_sqrt2_hi, h_phi_lo, sq_nonneg (Real.sqrt 2),
  83               mul_pos (show Real.sqrt 2 > 0 by positivity) (show phi > 0 from phi_pos)]
  84  -- log(phi) < 1 since phi < e
  85  have h_log_phi_lt1 : Real.log phi < 1 := by
  86    have h_e : Real.exp 1 > phi := by
  87      have h_exp1_ge2 : Real.exp 1 ≥ 2 := by
  88        have := Real.add_one_le_exp (1 : ℝ)
  89        linarith
  90      linarith [phi_lt_onePointSixTwo]
  91    rw [← Real.log_exp 1]
  92    exact Real.log_lt_log phi_pos h_e
  93  -- f_gap = w8 * log(phi) < w8 < 5 < alpha_seed - 2
  94  have h_fgap_lt_w8 : f_gap < w8_from_eight_tick := by
  95    unfold f_gap
  96    calc w8_from_eight_tick * Real.log phi
  97        < w8_from_eight_tick * 1 :=
  98          mul_lt_mul_of_pos_left h_log_phi_lt1 w8_pos
  99      _ = w8_from_eight_tick := mul_one _
 100  have h_fgap_small : f_gap < alpha_seed - 2 := by
 101    calc f_gap < w8_from_eight_tick := h_fgap_lt_w8
 102      _ < 5 := h_w8_lt5
 103      _ < alpha_seed - 2 := by linarith
 104  -- alphaInv ≥ alpha_seed - f_gap > 2
 105  calc alphaInv = alpha_seed * Real.exp (-(f_gap / alpha_seed)) := rfl
 106    _ ≥ alpha_seed * (1 - f_gap / alpha_seed) :=
 107        mul_le_mul_of_nonneg_left h_exp_ge (le_of_lt h_seed_pos)
 108    _ = alpha_seed - f_gap := by
 109        have h : alpha_seed ≠ 0 := ne_of_gt h_seed_pos
 110        field_simp
 111    _ > 2 := by linarith
 112
 113/-- **CALCULATED**: Ω_Λ > 0 (since α/π < 11/16 ≈ 0.6875).
 114
 115    Since alphaInv > 2, alpha = 1/alphaInv < 1/2.
 116    And alpha/pi < alpha (since pi > 1) < 1/2 < 11/16. -/
 117theorem omega_lambda_positive : 11/16 - (alpha / Real.pi) > 0 := by
 118  have h_alphaInv_pos : alphaInv > 0 := by unfold alphaInv alpha_seed; positivity
 119  have h_alpha_pos : alpha > 0 := by unfold alpha; positivity
 120  -- alpha < 1/2 since alphaInv > 2
 121  have h_alpha_lt_half : alpha < 1/2 := by
 122    have h_eq : alpha = 1/alphaInv := by unfold alpha; field_simp
 123    rw [h_eq]
 124    rw [div_lt_div_iff₀ h_alphaInv_pos (by norm_num : (0:ℝ) < 2)]
 125    linarith [alphaInv_gt_2]
 126  -- alpha/pi < alpha (since pi > 1)
 127  have h_pi_gt_1 : Real.pi > 1 := by linarith [Real.pi_gt_three]
 128  have h_ratio : alpha / Real.pi < alpha := div_lt_self h_alpha_pos h_pi_gt_1
 129  linarith
 130
 131/-- **BOUNDS**: 0 < Ω_Λ < 11/16 -/
 132theorem omega_lambda_bounds : 0 < 11/16 - (alpha / Real.pi) ∧ 11/16 - (alpha / Real.pi) < 11/16 :=
 133  ⟨omega_lambda_positive, omega_lambda_lt_11_16⟩
 134
 135/-! ## Section P-002: Fermion Mass Hierarchy -/
 136
 137/-- **CALCULATED**: φ^6 bounds: 17 < φ^6 < 18.
 138    Uses φ^6 = (φ^3)^2 = (2φ+1)^2. -/
 139theorem phi_6_hierarchy_bounds : (17 : ℝ) < (phi : ℝ)^6 ∧ (phi : ℝ)^6 < (18 : ℝ) := by
 140  have h3 := phi_cubed_eq  -- phi^3 = 2*phi + 1
 141  have hphi_lo : phi > 1.61 := phi_gt_onePointSixOne
 142  have hphi_hi : phi < 1.62 := phi_lt_onePointSixTwo
 143  have h6 : phi^6 = (2 * phi + 1)^2 := by nlinarith [phi_pos]
 144  constructor
 145  · rw [h6]; nlinarith
 146  · rw [h6]; nlinarith
 147
 148/-- **CALCULATED**: φ^11 > 180 (conservative lower bound for large hierarchies).
 149    Uses Fibonacci: phi^11 = 89*phi + 55 > 89*1.618 + 55 > 180. -/
 150theorem phi_11_hierarchy_lower : (180 : ℝ) < (phi : ℝ)^11 := by
 151  rw [phi_eleventh_eq]
 152  linarith [phi_gt_onePointSixOne]
 153
 154/-- **HIERARCHY STRUCTURE**: Mass ratios are φ-powers of integer differences. -/
 155theorem hierarchy_phi_power_structure (Δr : ℕ) (hΔr : Δr > 0) :
 156    ∃ (ratio : ℝ), ratio = (phi : ℝ)^Δr ∧ ratio > 1 := by
 157  use (phi : ℝ)^Δr
 158  refine ⟨rfl, ?_⟩
 159  have h1 : phi > 1 := one_lt_phi
 160  have h_ge : Δr ≥ 1 := hΔr
 161  calc 1 = 1^Δr := (one_pow Δr).symm
 162    _ < phi^Δr := by
 163        apply pow_lt_pow_left₀ h1 (by norm_num)
 164        exact Nat.pos_iff_ne_zero.mp hΔr
 165
 166/-! ## Section: Combined Registry Certificate -/
 167
 168/-- **CERTIFICATE**: Registry predictions with calculated bounds. -/
 169structure RegistryPredictionsCert where
 170  omega_lambda_upper : 11/16 - (alpha / Real.pi) < 11/16
 171  omega_lambda_lower : 0 < 11/16 - (alpha / Real.pi)
 172  phi_6_lower : (17 : ℝ) < (phi : ℝ)^6
 173  phi_6_upper : (phi : ℝ)^6 < (18 : ℝ)
 174  phi_11_lower : (180 : ℝ) < (phi : ℝ)^11
 175  hierarchy_exists : ∀ (Δr : ℕ), Δr > 0 →
 176    ∃ (ratio : ℝ), ratio = (phi : ℝ)^Δr ∧ ratio > 1
 177
 178/-- **THEOREM**: Registry predictions certificate is inhabited. -/
 179theorem registry_predictions_cert_exists : ∃ _ : RegistryPredictionsCert, True :=
 180  ⟨⟨omega_lambda_lt_11_16,
 181    omega_lambda_positive,
 182    phi_6_hierarchy_bounds.1,
 183    phi_6_hierarchy_bounds.2,
 184    phi_11_hierarchy_lower,
 185    fun Δr hΔr => hierarchy_phi_power_structure Δr hΔr⟩,
 186   trivial⟩
 187
 188end RegistryPredictionsProved
 189end Unification
 190end IndisputableMonolith
 191