IndisputableMonolith.Relativity.GRLimit.Parameters

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Parameter Limits and Recognition Spine Connection
   6
   7ACTUALLY PROVES that ILG parameters (α, C_lag) from RS are small and perturbative.
   8
   9From Source.txt line 26:
  10- α = (1 - 1/φ)/2 (derived from RS geometry)
  11- C_lag = φ^(-5) (derived from coherence quantum E_coh = φ^(-5) eV)
  12
  13We PROVE (not assume):
  141. Both < 1 (straightforward)
  152. Product < 0.1 (requires showing φ^5 > 10)
  163. Product < 0.02 (STATUS: needs tighter bounds - see below)
  17-/
  18
  19namespace IndisputableMonolith
  20namespace Relativity
  21namespace GRLimit
  22
  23/-- ILG exponent α from RS: α = (1 - 1/φ)/2 ≈ 0.191 -/
  24noncomputable def alpha_from_phi : ℝ :=
  25  (1 - 1 / Constants.phi) / 2
  26
  27/-- ILG lag constant C_lag from RS: C_lag = φ^(-5) ≈ 0.090 -/
  28noncomputable def cLag_from_phi : ℝ :=
  29  Constants.phi ^ (-5 : ℝ)
  30
  31/-- PROVEN: Both parameters are positive. -/
  32theorem rs_params_positive :
  33  0 < alpha_from_phi ∧ 0 < cLag_from_phi := by
  34  constructor
  35  · unfold alpha_from_phi
  36    have hφ_pos : 0 < Constants.phi := Constants.phi_pos
  37    have hφ_gt_one : 1 < Constants.phi := Constants.one_lt_phi
  38    have : 0 < 1 - 1 / Constants.phi := by
  39      have : 1 / Constants.phi < 1 := (div_lt_one hφ_pos).mpr hφ_gt_one
  40      linarith
  41    linarith
  42  · unfold cLag_from_phi
  43    exact Real.rpow_pos_of_pos Constants.phi_pos _
  44
  45/-- PROVEN: α < 1 (straightforward from φ > 1). -/
  46theorem alpha_lt_one : alpha_from_phi < 1 := by
  47  unfold alpha_from_phi
  48  have hφ_pos : 0 < Constants.phi := Constants.phi_pos
  49  have : 1 - 1 / Constants.phi < 1 := by
  50    have : 0 < 1 / Constants.phi := div_pos (by norm_num) hφ_pos
  51    linarith
  52  have : (1 - 1 / Constants.phi) / 2 < 1 / 2 := by
  53    exact div_lt_div_of_pos_right this (by norm_num)
  54  linarith
  55
  56/- PROVEN: α < 1/2 (since 1 − 1/φ < 1). -/
  57theorem alpha_lt_half : alpha_from_phi < 1 / 2 := by
  58  unfold alpha_from_phi
  59  have hφ_pos : 0 < Constants.phi := Constants.phi_pos
  60  have : 1 - 1 / Constants.phi < 1 := by
  61    have : 0 < 1 / Constants.phi := div_pos (by norm_num) hφ_pos
  62    linarith
  63  exact div_lt_div_of_pos_right this (by norm_num)
  64
  65-- (helper lemma removed)
  66
  67/-- φ > 3/2. -/
  68theorem phi_gt_three_halves : Constants.phi > 3 / 2 := by
  69  -- First show √5 > 11/5, hence φ = (1+√5)/2 > (1+11/5)/2 = 8/5 > 3/2
  70  have hy : 0 ≤ (11 : ℝ) / 5 := by norm_num
  71  have hnot_le : ¬ (Real.sqrt 5 ≤ (11 : ℝ) / 5) := by
  72    -- If √5 ≤ 11/5 then 5 ≤ (11/5)^2, contradiction
  73    have hcontra : ¬ (5 : ℝ) ≤ ((11 : ℝ) / 5) ^ 2 := by norm_num
  74    exact fun hle => hcontra ((Real.sqrt_le_left hy).mp hle)
  75  have h11lt : (11 : ℝ) / 5 < Real.sqrt 5 := lt_of_not_ge hnot_le
  76  have hsum : 1 + (11 : ℝ) / 5 < 1 + Real.sqrt 5 := by linarith
  77  have hdiv : (1 + (11 : ℝ) / 5) / 2 < (1 + Real.sqrt 5) / 2 :=
  78    div_lt_div_of_pos_right hsum (by norm_num)
  79  have h8over5 : (8 : ℝ) / 5 = (1 + (11 : ℝ) / 5) / 2 := by norm_num
  80  have hphi : (1 + Real.sqrt 5) / 2 = Constants.phi := by simp [Constants.phi]
  81  have h8ltphi : (8 : ℝ) / 5 < Constants.phi := by
  82    simpa [h8over5, hphi] using hdiv
  83  have : (3 : ℝ) / 2 < (8 : ℝ) / 5 := by norm_num
  84  exact lt_trans this h8ltphi
  85
  86-- φ^2 = φ + 1 (reference)
  87
  88-- φ^5 = 5φ + 3 (reference)
  89
  90-- φ^5 > 10 (reference); not needed since we bound C_lag via rpow monotonicity
  91
  92/-- PROVEN: φ^(-5) < 1/10. -/
  93theorem cLag_lt_one_tenth : cLag_from_phi < 1 / 10 := by
  94  -- Use φ ≥ 8/5 and negative exponent monotonicity: φ^(−5) ≤ (8/5)^(−5) = 3125/32768 < 1/10
  95  unfold cLag_from_phi
  96  have hphi_ge : (8 : ℝ) / 5 ≤ Constants.phi := le_of_lt (by
  97    -- φ > 8/5
  98    unfold Constants.phi
  99    have hy : 0 ≤ (11 : ℝ) / 5 := by norm_num
 100    have hnot_le : ¬ (Real.sqrt 5 ≤ (11 : ℝ) / 5) := by
 101      have hcontra : ¬ (5 : ℝ) ≤ ((11 : ℝ) / 5) ^ 2 := by norm_num
 102      exact fun hle => hcontra ((Real.sqrt_le_left hy).mp hle)
 103    have h11lt : (11 : ℝ) / 5 < Real.sqrt 5 := lt_of_not_ge hnot_le
 104    have hsum : 1 + (11 : ℝ) / 5 < 1 + Real.sqrt 5 := by linarith
 105    have hdiv : (1 + (11 : ℝ) / 5) / 2 < (1 + Real.sqrt 5) / 2 :=
 106      div_lt_div_of_pos_right hsum (by norm_num)
 107    have h8over5 : (8 : ℝ) / 5 = (1 + (11 : ℝ) / 5) / 2 := by norm_num
 108    have hphi : (1 + Real.sqrt 5) / 2 = Constants.phi := by simp [Constants.phi]
 109    simpa [h8over5, hphi] using hdiv)
 110  have hxpos : 0 < (8 : ℝ) / 5 := by norm_num
 111  have hmon : Constants.phi ^ ((-5) : ℝ) ≤ ((8 : ℝ) / 5) ^ ((-5) : ℝ) :=
 112    Real.rpow_le_rpow_of_nonpos hxpos hphi_ge (by norm_num)
 113  have hrpow : ((8 : ℝ) / 5) ^ ((-5) : ℝ) = 1 / ((8 : ℝ) / 5) ^ 5 := by
 114    rw [Real.rpow_neg (le_of_lt hxpos)]
 115    simp
 116  have hlt : 1 / ((8 : ℝ) / 5) ^ 5 < 1 / 10 := by
 117    -- since (8/5)^5 = 32768/3125 > 10
 118    have hpow : ((8 : ℝ) / 5) ^ 5 = (32768 : ℝ) / 3125 := by norm_num
 119    have hgt : ((8 : ℝ) / 5) ^ 5 > 10 := by simpa [hpow] using (by norm_num : (32768 : ℝ) / 3125 > 10)
 120    exact (div_lt_div_of_pos_left (by norm_num) (by norm_num) hgt)
 121  have : ((8 : ℝ) / 5) ^ ((-5) : ℝ) < 1 / 10 := by simpa [hrpow] using hlt
 122  exact lt_of_le_of_lt hmon this
 123
 124/-- PROVEN: C_lag < 1 (from φ^5 > 10 ⇒ φ^(−5) < 1/10 < 1). -/
 125theorem cLag_lt_one : cLag_from_phi < 1 := by
 126  have hlt : cLag_from_phi < 1 / 10 := cLag_lt_one_tenth
 127  have : (1 / 10 : ℝ) < 1 := by norm_num
 128  exact lt_trans hlt this
 129
 130/-- PROVEN: Product < 0.1 using algebraic bounds. -/
 131theorem rs_params_perturbative_proven : |alpha_from_phi * cLag_from_phi| < 0.1 := by
 132  have hα_pos := rs_params_positive.1
 133  have hC_pos := rs_params_positive.2
 134  rw [abs_of_nonneg (mul_nonneg (le_of_lt hα_pos) (le_of_lt hC_pos))]
 135  have hα_lt : alpha_from_phi < 1 / 2 := alpha_lt_half
 136  have hC_lt : cLag_from_phi < 1 / 10 := cLag_lt_one_tenth
 137  calc alpha_from_phi * cLag_from_phi
 138      < (1 / 2) * (1 / 10) := by
 139        apply mul_lt_mul'' hα_lt hC_lt (le_of_lt hα_pos) (le_of_lt hC_pos)
 140    _ = 1 / 20 := by norm_num
 141    _ < 0.1 := by norm_num
 142
 143/-- STATUS: Product < 0.02 needs tighter bounds.
 144
 145    PROGRESS: Proven product < 0.05 (since α < 1/2, C_lag < 1/10)
 146    NEEDED: Either α < 1/5 OR C_lag < 1/11 to get product < 0.02
 147
 148    Current bounds:
 149    - α = (1-1/φ)/2 where φ = (1+√5)/2
 150    - Need to show α < 1/5 OR find tighter C_lag bound
 151
 152    Path forward:
 153    - Prove φ < 1.62 ⟹ 1/φ > 0.617 ⟹ 1-1/φ < 0.383 ⟹ α < 0.192 < 1/5 ✓
 154    - Requires proving √5 < 2.24 ⟹ φ < (1+2.24)/2 = 1.62
 155    - This is doable with Mathlib's Real.sqrt inequalities
 156-/
 157theorem coupling_product_small_proven : |alpha_from_phi * cLag_from_phi| < 0.02 := by
 158  have hα_pos := rs_params_positive.1
 159  have hC_pos := rs_params_positive.2
 160  rw [abs_of_nonneg (mul_nonneg (le_of_lt hα_pos) (le_of_lt hC_pos))]
 161
 162  -- Strategy: Prove α < 1/5
 163  -- Need: (1 - 1/φ)/2 < 1/5
 164  -- ⟺ 1 - 1/φ < 2/5
 165  -- ⟺ 1 - 2/5 < 1/φ
 166  -- ⟺ 3/5 < 1/φ
 167  -- ⟺ φ < 5/3
 168
 169  have hα_lt_one_fifth : alpha_from_phi < 1 / 5 := by
 170    unfold alpha_from_phi
 171    have hφ_pos : 0 < Constants.phi := Constants.phi_pos
 172
 173    -- Need to prove φ < 5/3
 174    have hφ_lt_5_3 : Constants.phi < 5 / 3 := by
 175      unfold Constants.phi
 176      -- (1+√5)/2 < 5/3
 177      -- ⟺ 3(1+√5) < 10
 178      -- ⟺ 3 + 3√5 < 10
 179      -- ⟺ 3√5 < 7
 180      -- ⟺ √5 < 7/3
 181      -- ⟺ 5 < 49/9
 182      -- 5 = 45/9 < 49/9 ✓
 183      have h_sqrt5_lt : Real.sqrt 5 < 7 / 3 := by
 184        -- use sqrt_lt equivalence: √x < y ↔ x < y^2
 185        have hx : 0 ≤ (5 : ℝ) := by norm_num
 186        have hy : 0 ≤ (7 / 3 : ℝ) := by norm_num
 187        have hxy : (5 : ℝ) < (7 / 3 : ℝ) ^ 2 := by norm_num
 188        exact (Real.sqrt_lt hx hy).2 hxy
 189      have : 1 + Real.sqrt 5 < 1 + 7 / 3 := by linarith
 190      have : (1 + Real.sqrt 5) / 2 < (1 + 7 / 3) / 2 := by
 191        exact div_lt_div_of_pos_right this (by norm_num)
 192      calc (1 + Real.sqrt 5) / 2
 193          < (1 + 7 / 3) / 2 := this
 194        _ = 10 / 6 := by norm_num
 195        _ = 5 / 3 := by norm_num
 196
 197    -- Now: φ < 5/3 ⟹ 1/φ > 3/5 ⟹ 1 - 1/φ < 2/5 ⟹ α < 1/5
 198    have : 1 / Constants.phi > 3 / 5 := by
 199      -- From φ < 5/3 and φ > 0, we get 1/(5/3) < 1/φ i.e., 3/5 < 1/φ
 200      have hpos : 0 < Constants.phi := hφ_pos
 201      have : 1 / (5 / 3 : ℝ) < 1 / Constants.phi :=
 202        one_div_lt_one_div_of_lt hpos hφ_lt_5_3
 203      simpa using this
 204    have : 1 - 1 / Constants.phi < 2 / 5 := by linarith
 205    have : (1 - 1 / Constants.phi) / 2 < (2 / 5) / 2 := by
 206      exact div_lt_div_of_pos_right this (by norm_num)
 207    calc (1 - 1 / Constants.phi) / 2
 208        < (2 / 5) / 2 := this
 209      _ = 1 / 5 := by norm_num
 210
 211  have hC_lt : cLag_from_phi < 1 / 10 := cLag_lt_one_tenth
 212
 213  calc alpha_from_phi * cLag_from_phi
 214      < (1 / 5) * (1 / 10) := by
 215        apply mul_lt_mul'' hα_lt_one_fifth hC_lt (le_of_lt hα_pos) (le_of_lt hC_pos)
 216    _ = 1 / 50 := by norm_num
 217    _ = 0.02 := by norm_num
 218
 219/-- PROVEN: Both parameters < 1. -/
 220theorem rs_params_small_proven : alpha_from_phi < 1 ∧ cLag_from_phi < 1 :=
 221  ⟨alpha_lt_one, cLag_lt_one⟩
 222
 223/-- Recognition spine parameters are small (for perturbation theory). -/
 224class GRLimitParameterFacts : Prop where
 225  rs_params_small : alpha_from_phi < 1 ∧ cLag_from_phi < 1
 226  coupling_product_small : |alpha_from_phi * cLag_from_phi| < 0.02
 227  rs_params_perturbative : (|alpha_from_phi * cLag_from_phi|) < 0.1
 228
 229/-- Rigorous instance providing GRLimitParameterFacts with ACTUAL PROOFS. -/
 230instance grLimitParameterFacts_proven : GRLimitParameterFacts where
 231  rs_params_small := rs_params_small_proven
 232  coupling_product_small := coupling_product_small_proven
 233  rs_params_perturbative := rs_params_perturbative_proven
 234
 235end GRLimit
 236end Relativity
 237end IndisputableMonolith
 238