IndisputableMonolith.Gravity.RunningG

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# C51: Gravitational Running at Nanometer Scales
   6
   7This module formalizes the prediction that Newton's gravitational constant G
   8is not truly constant, but "runs" (strengthens) at nanometer scales.
   9
  10## The Theory
  11
  121. **Macroscopic Limit**: G(r) -> G_∞ as r -> ∞.
  132. **Nanoscale Enhancement**: At r ≈ 20 nm, G(r) ≈ 32 * G_∞.
  143. **Running Exponent**: The deviation follows an exponent β derived from the φ-ladder.
  15   β = -(φ - 1) / φ^5 ≈ -0.056.
  16
  17## Prediction
  18
  19The effective gravitational constant G_eff(r) follows:
  20  G_eff(r) = G_∞ * (1 + |β| * (r / r_ref)^β)
  21where r_ref is the scale at which the correction becomes order unity.
  22-/
  23
  24namespace IndisputableMonolith
  25namespace Gravity
  26namespace RunningG
  27
  28open Constants
  29
  30/-- The running exponent for gravitational strengthening.
  31    β = -(φ - 1) / φ^5 ≈ -0.056. -/
  32noncomputable def beta_running : ℝ := -(phi - 1) / (phi ^ 5)
  33
  34/-- Numerical bound for beta_running ≈ -0.0557.
  35    Proved using φ ∈ (1.61, 1.62). -/
  36theorem beta_running_bounds :
  37    -0.06 < beta_running ∧ beta_running < -0.05 := by
  38  unfold beta_running
  39  -- Use phi_fifth_eq: φ^5 = 5φ + 3
  40  rw [phi_fifth_eq]
  41  -- We want to prove: -0.06 < -(φ - 1) / (5φ + 3) < -0.05
  42  have h_phi_pos : 0 < phi := phi_pos
  43  have h_denom_pos : 0 < 5 * phi + 3 := by linarith
  44  constructor
  45  · -- -0.06 < -(φ - 1) / (5φ + 3)
  46    rw [lt_div_iff₀ h_denom_pos]
  47    have h_phi_lt : phi < 1.62 := phi_lt_onePointSixTwo
  48    linarith
  49  · -- -(φ - 1) / (5φ + 3) < -0.05
  50    rw [div_lt_iff₀ h_denom_pos]
  51    have h_phi_gt : 1.61 < phi := phi_gt_onePointSixOne
  52    linarith
  53
  54/-- Effective G at scale r relative to G_infinity. -/
  55noncomputable def G_ratio (r r_ref : ℝ) : ℝ :=
  56    1 + abs beta_running * (r / r_ref) ^ beta_running
  57
  58/-- **HYPOTHESIS H_GravitationalRunning**: Gravity strengthens at nm scales.
  59    Prediction: G(20nm) / G_inf ≈ 32. -/
  60def H_GravitationalRunning : Prop :=
  61  ∃ r_ref : ℝ, r_ref > 0 ∧
  62    abs (G_ratio 20e-9 r_ref - 32) < 1.0
  63
  64/-! ## Structural Properties of G_ratio -/
  65
  66/-- beta_running is strictly negative. -/
  67theorem beta_running_neg : beta_running < 0 := by
  68  have := beta_running_bounds
  69  linarith [this.2]
  70
  71/-- |beta_running| is strictly positive. -/
  72theorem abs_beta_running_pos : 0 < abs beta_running := by
  73  exact abs_pos.mpr (ne_of_lt beta_running_neg)
  74
  75/-- At r_ref = r, G_ratio(r, r) = 1 + |β|.
  76    The base (r/r) = 1, and 1^β = 1 for any β. -/
  77theorem G_ratio_at_self (r : ℝ) (hr : 0 < r) :
  78    G_ratio r r = 1 + abs beta_running := by
  79  unfold G_ratio
  80  rw [div_self (ne_of_gt hr), Real.one_rpow]
  81  ring
  82
  83/-- G_ratio at r_ref = r is less than 2 (and hence far below 31).
  84    Since |β| < 0.06 < 1, we have 1 + |β| < 2. -/
  85theorem G_ratio_at_self_lt_two (r : ℝ) (hr : 0 < r) :
  86    G_ratio r r < 2 := by
  87  rw [G_ratio_at_self r hr]
  88  have hbeta := beta_running_bounds
  89  have h_abs : abs beta_running < 0.06 := by
  90    rw [abs_of_neg beta_running_neg]
  91    linarith [hbeta.1]
  92  linarith
  93
  94/-- G_ratio at r_ref = r is less than 31 (needed for IVT with target 32). -/
  95theorem G_ratio_at_self_lt_31 (r : ℝ) (hr : 0 < r) :
  96    G_ratio r r < 31 := by
  97  have := G_ratio_at_self_lt_two r hr
  98  linarith
  99
 100/-- G_ratio at r_ref = r is positive (it equals 1 + |beta| > 1). -/
 101theorem G_ratio_at_self_pos (r : ℝ) (hr : 0 < r) : 0 < G_ratio r r := by
 102  rw [G_ratio_at_self r hr]; linarith [abs_beta_running_pos]
 103
 104/-! ## Monotonicity and Unboundedness of G_ratio -/
 105
 106/-- G_ratio is monotonically increasing in r_ref (for fixed r > 0 and beta < 0).
 107    As r_ref grows, (r/r_ref) shrinks, and raising a number in (0,1) to a
 108    negative power gives a LARGER result. -/
 109theorem G_ratio_mono (r : ℝ) (hr : 0 < r) (R1 R2 : ℝ)
 110    (hR1 : 0 < R1) (hR12 : R1 ≤ R2) :
 111    G_ratio r R1 ≤ G_ratio r R2 := by
 112  unfold G_ratio
 113  have hab : 0 < abs beta_running := abs_beta_running_pos
 114  have hbeta_neg : beta_running < 0 := beta_running_neg
 115  suffices h : (r / R1) ^ beta_running ≤ (r / R2) ^ beta_running by
 116    linarith [mul_le_mul_of_nonneg_left h (le_of_lt hab)]
 117  apply Real.rpow_le_rpow_of_exponent_le
 118  · exact le_of_lt (div_pos hr hR1)
 119  · exact div_le_div_of_nonneg_left hr hR1 hR12
 120  · exact le_of_lt hbeta_neg
 121
 122/-- For any target M, there exists R > r with G_ratio(r, R) > M.
 123    G_ratio grows without bound as r_ref increases. -/
 124theorem G_ratio_eventually_large (r : ℝ) (hr : 0 < r) (M : ℝ) :
 125    ∃ R : ℝ, R > r ∧ M < G_ratio r R := by
 126  by_cases hM : M < G_ratio r r
 127  · exact ⟨r, lt_irrefl r |>.elim ▸ ⟨lt_of_le_of_lt (le_refl r) (by linarith), hM⟩⟩
 128  · push_neg at hM
 129    have hM_bound : G_ratio r r ≤ M := hM
 130    use r + M + 1
 131    refine ⟨by linarith, ?_⟩
 132    calc M < G_ratio r r + (M - G_ratio r r + 1) := by linarith
 133      _ ≤ G_ratio r (r + M + 1) + (M - G_ratio r r + 1) := by
 134          linarith [G_ratio_mono r hr r (r + M + 1) hr (by linarith : r ≤ r + M + 1)]
 135      _ = G_ratio r (r + M + 1) + (M - G_ratio r r + 1) := rfl
 136
 137/-- G_ratio is continuous in r_ref on (0, infinity). -/
 138theorem G_ratio_continuous_snd (r : ℝ) (hr : 0 < r) :
 139    ContinuousOn (G_ratio r) (Set.Ioi 0) := by
 140  unfold G_ratio
 141  apply ContinuousOn.add continuousOn_const
 142  apply ContinuousOn.mul continuousOn_const
 143  apply ContinuousOn.rpow_const
 144  · exact ContinuousOn.div continuousOn_const continuousOn_id (fun x hx => ne_of_gt hx)
 145  · exact fun x hx => Or.inl (ne_of_gt (div_pos hr hx))
 146
 147/-- **EXISTENCE THEOREM**: The 20nm gravity prediction is satisfiable.
 148    There exists r_ref > 0 with |G_ratio(20nm, r_ref) - 32| < 1. -/
 149theorem H_GravitationalRunning_certificate : H_GravitationalRunning := by
 150  unfold H_GravitationalRunning
 151  have hr : (0 : ℝ) < 20e-9 := by norm_num
 152  have h_small := G_ratio_at_self_lt_31 20e-9 hr
 153  obtain ⟨R, hRr, hR_large⟩ := G_ratio_eventually_large 20e-9 hr 33
 154  have hR_pos : 0 < R := lt_trans hr hRr
 155  have h_cont := G_ratio_continuous_snd 20e-9 hr
 156  have h_sub : Set.Icc (20e-9 : ℝ) R ⊆ Set.Ioi 0 :=
 157    fun x ⟨hlo, _⟩ => lt_of_lt_of_le hr hlo
 158  have h_ivt := intermediate_value_Icc (le_of_lt hRr) (h_cont.mono h_sub)
 159  have h32 : (32 : ℝ) ∈ Set.Icc (G_ratio 20e-9 20e-9) (G_ratio 20e-9 R) :=
 160    ⟨by linarith, by linarith⟩
 161  obtain ⟨c, ⟨hc_lo, _⟩, hc_eq⟩ := h_ivt h32
 162  exact ⟨c, lt_of_lt_of_le hr hc_lo, by rw [hc_eq]; simp⟩
 163
 164/-! ## Q9: Is r_ref Derivable from phi?
 165
 166**Analysis**: beta = -(phi-1)/phi^5 is derived from phi. But r_ref (the
 167scale at which running G reaches a particular enhancement) is determined
 168by the IVT -- its value is NOT constrained by the forcing chain alone.
 169
 170**Current status**: r_ref is a free parameter. Deriving it would require
 171either the Fibonacci-square conjecture (N_tau = F_12 - 2 = 142) from
 172GravityParameters.lean, or empirical input from short-range experiments. -/
 173
 174/-- The hypothesis that r_ref lives on the phi-ladder. -/
 175def H_rref_phi_ladder : Prop :=
 176  ∃ N : ℤ, ∃ r_ref : ℝ, r_ref = ell0 * phi ^ N ∧ r_ref > 0 ∧
 177    abs (G_ratio 20e-9 r_ref - 32) < 1
 178
 179/-! ## Q10: Casimir Force Correction
 180
 181Running G at 20nm gives G_eff ≈ 32 * G_inf. But G_inf ≈ 6.7e-11 makes
 182even the enhanced gravitational force negligible vs Casimir (~10 Pa at 20nm).
 183The fractional gravitational correction to Casimir is ≈ 2e-18. -/
 184
 185/-- Gravitational pressure between two plates. -/
 186def gravitational_pressure (G_val rho t enhancement : ℝ) : ℝ :=
 187  enhancement * G_val * rho ^ 2 * t ^ 2
 188
 189/-- The gravitational contribution is negligibly small vs Casimir. -/
 190theorem grav_casimir_ratio_negligible :
 191    gravitational_pressure 6.674e-11 1e4 1e-6 32 < 1e-10 := by
 192  unfold gravitational_pressure; norm_num
 193
 194/-! ## Explicit r_ref Formula (Path 1a)
 195
 196Setting G_ratio(r, r_ref) = target and solving for r_ref:
 197  target = 1 + |beta| * (r / r_ref)^beta
 198  (target - 1) / |beta| = (r / r_ref)^beta
 199  r_ref = r * ((target - 1) / |beta|)^(1/beta)
 200
 201Since beta < 0, the exponent 1/beta < 0, and (target-1)/|beta| > 1 for
 202target > 1 + |beta|, so r_ref > r (the reference scale is larger than
 203the measurement scale). -/
 204
 205/-- The explicit r_ref that gives G_ratio(r, r_ref) = target.
 206    Derived by inverting the G_ratio formula. -/
 207noncomputable def r_ref_exact (r target : ℝ) : ℝ :=
 208  r * ((target - 1) / abs beta_running) ^ (1 / beta_running)
 209
 210/-- r_ref_exact is positive when r > 0 and target > 1. -/
 211theorem r_ref_exact_pos (r target : ℝ) (hr : 0 < r) (ht : 1 < target) :
 212    0 < r_ref_exact r target := by
 213  unfold r_ref_exact
 214  apply mul_pos hr
 215  apply Real.rpow_pos_of_pos
 216  exact div_pos (by linarith) abs_beta_running_pos
 217
 218/-- For target > 1 + |beta| (i.e., target above the G_ratio at self),
 219    r_ref_exact > r. The reference scale is LARGER than the measurement scale. -/
 220theorem r_ref_exact_gt_r (r target : ℝ) (hr : 0 < r)
 221    (ht : 1 + abs beta_running < target) :
 222    r < r_ref_exact r target := by
 223  unfold r_ref_exact
 224  have h_base_gt_one : 1 < (target - 1) / abs beta_running := by
 225    rw [one_lt_div abs_beta_running_pos]; linarith
 226  have h_exp_neg : 1 / beta_running < 0 := by
 227    apply div_neg_of_pos_of_neg one_pos beta_running_neg
 228  have h_rpow_pos : 0 < ((target - 1) / abs beta_running) ^ (1 / beta_running) :=
 229    Real.rpow_pos_of_pos (lt_trans one_pos h_base_gt_one) _
 230  calc r = r * 1 := by ring
 231    _ < r * ((target - 1) / abs beta_running) ^ (1 / beta_running) := by
 232        apply mul_lt_mul_of_pos_left _ hr
 233        exact Real.one_lt_rpow h_base_gt_one.le h_exp_neg
 234
 235/-! ## Phi-Ladder Rung Analysis (Path 1b)
 236
 237For target = 32, r = 20 nm:
 238  r_ref = 20e-9 * (31/|beta|)^(1/beta)
 239  |beta| ~ 0.0557, 1/beta ~ -17.95
 240  31/0.0557 ~ 556.6
 241  556.6^(-17.95) ~ 1.83e49
 242  r_ref ~ 20e-9 * 1.83e49 ~ 3.66e41 m
 243
 244In Planck units (ell_P ~ 1.6e-35 m):
 245  r_ref / ell_P ~ 2.3e76
 246  log_phi(2.3e76) ~ 76 * ln(10) / ln(phi) ~ 76 * 2.303 / 0.481 ~ 364
 247
 248So r_ref sits near phi-rung N ~ 364.
 249
 250Significance: 364 = 4 * 91 = 4 * 7 * 13.
 251Also: 364 = F_14 - 13 (where F_14 = 377).
 252And: 364 = 8 * 45 + 4 = 8 * 45.5 (close to 8 * gap_45 = 360).
 253
 254The nearest "clean" RS number is 360 = lcm(8, 45) = sync_period from
 255Foundation.DimensionForcing. So r_ref ~ ell_P * phi^360 is suggestive. -/
 256
 257/-- The approximate phi-rung of r_ref for the 20nm/32x prediction. -/
 258def r_ref_phi_rung_approx : ℕ := 364
 259
 260/-- 364 is close to 360 = lcm(8, 45) = the RS sync period. -/
 261theorem rung_near_sync_period : r_ref_phi_rung_approx - 360 = 4 := by
 262  native_decide
 263
 264/-- 360 = 8 * 45 (8-tick times gap-45). -/
 265theorem sync_period_factored : 360 = 8 * 45 := by norm_num
 266
 267/-- If r_ref = ell0 * phi^360, the prediction is tied to the sync period
 268    from D=3 forcing. This makes r_ref a zero-parameter consequence of
 269    the forcing chain (modulo the 4-rung offset). -/
 270def H_rref_sync_period : Prop :=
 271  ∃ r_ref : ℝ, r_ref = ell0 * phi ^ (360 : ℝ) ∧ r_ref > 0 ∧
 272    abs (G_ratio 20e-9 r_ref - 32) < 2
 273
 274/-- Running G Predictions Certificate (Round 4). -/
 275structure RunningGR4Cert where
 276  r_ref_explicit : ∀ r target : ℝ, 0 < r → 1 < target → 0 < r_ref_exact r target
 277  r_ref_above_r : ∀ r target : ℝ, 0 < r → 1 + abs beta_running < target →
 278    r < r_ref_exact r target
 279  rung_near_360 : r_ref_phi_rung_approx - 360 = 4
 280  hypothesis_exists : H_GravitationalRunning
 281
 282theorem running_g_r4_cert : RunningGR4Cert where
 283  r_ref_explicit := r_ref_exact_pos
 284  r_ref_above_r := r_ref_exact_gt_r
 285  rung_near_360 := rung_near_sync_period
 286  hypothesis_exists := H_GravitationalRunning_certificate
 287
 288end RunningG
 289end Gravity
 290end IndisputableMonolith
 291