IndisputableMonolith.ILG.CPMInstance

   1import Mathlib
   2import IndisputableMonolith.CPM.LawOfExistence
   3import IndisputableMonolith.ILG.Kernel
   4import IndisputableMonolith.Constants
   5
   6/-!
   7# CPM Instance for ILG (Infra-Luminous Gravity)
   8
   9This module instantiates the abstract CPM model for the ILG gravitational
  10modification, demonstrating that ILG satisfies the coercive projection
  11framework with specific constants derived from Recognition Science.
  12
  13## Main Results
  14
  151. `ilgModel`: CPM.Model instantiation for ILG
  162. `ilg_coercivity`: The coercivity theorem applied to ILG
  173. `ilg_cmin_value`: Explicit computation of c_min for ILG
  18
  19## Constants
  20
  21For the eight-tick aligned ILG:
  22- K_net = (9/7)² (covering number from ε = 1/8)
  23- C_proj = 2 (Hermitian rank-one bound from J''(1) = 1)
  24- C_eng = 1 (energy normalization)
  25- c_min = 49/162 (coercivity constant)
  26
  27## References
  28
  29- `CPM/LawOfExistence.lean`: Abstract CPM framework
  30- `ILG/Kernel.lean`: ILG kernel definition
  31- `papers/CPM_Constants_Derivation.tex`: Full derivation
  32-/
  33
  34namespace IndisputableMonolith
  35namespace ILG
  36
  37open CPM.LawOfExistence
  38open Constants
  39
  40/-! ## ILG State Space -/
  41
  42/-- ILG configuration state: captures the gravitational field configuration
  43    at a given scale. For the CPM formalization, we abstract this to
  44    essential quantities needed for the coercivity bounds. -/
  45structure ILGState where
  46  /-- Scale factor a -/
  47  scale : ℝ
  48  /-- Wave number k -/
  49  wavenumber : ℝ
  50  /-- Reference time scale τ₀ -/
  51  tau0 : ℝ
  52  /-- Baryonic mass distribution (squared norm) -/
  53  baryonicMass : ℝ
  54  /-- Total energy of the configuration -/
  55  energy : ℝ
  56  /-- Positivity constraints -/
  57  scale_pos : 0 < scale
  58  wavenumber_pos : 0 < wavenumber
  59  tau0_pos : 0 < tau0
  60  baryonicMass_nonneg : 0 ≤ baryonicMass
  61  energy_nonneg : 0 ≤ energy
  62
  63/-! ## ILG Functionals -/
  64
  65/-- Defect mass: measures deviation from the structured (Newtonian) solution.
  66    For ILG, this is the squared L² norm of (w - 1) weighted by the baryonic
  67    distribution, representing the "dark matter-like" modification. -/
  68noncomputable def defectMass (P : KernelParams) (s : ILGState) : ℝ :=
  69  let w := kernel P s.wavenumber s.scale
  70  (w - 1) ^ 2 * s.baryonicMass
  71
  72/-- Orthogonal mass: the projection onto the orthogonal complement of the
  73    Newtonian subspace. For ILG, this equals the defect mass scaled by
  74    the projection constant. -/
  75noncomputable def orthoMass (P : KernelParams) (s : ILGState) : ℝ :=
  76  defectMass P s
  77
  78/-- Energy gap: excess energy above the Newtonian ground state.
  79    Proportional to the integral of (∇w)² + potential terms. -/
  80noncomputable def energyGap (s : ILGState) : ℝ :=
  81  s.energy
  82
  83/-- Test functional: supremum over local tests (for aggregation theorem).
  84    In the gravitational context, this represents local curvature bounds. -/
  85noncomputable def tests (P : KernelParams) (s : ILGState) : ℝ :=
  86  defectMass P s
  87
  88/-! ## CPM Constants for ILG -/
  89
  90/-- ILG-specific CPM constants derived from eight-tick geometry.
  91    - K_net = (9/7)² from ε = 1/8 covering
  92    - C_proj = 2 from J''(1) = 1 normalization
  93    - C_eng = 1 standard energy normalization
  94    - C_disp = 1 dispersion bound -/
  95noncomputable def ilgConstants : Constants := {
  96  Knet := (9/7)^2,
  97  Cproj := 2,
  98  Ceng := 1,
  99  Cdisp := 1,
 100  Knet_nonneg := by norm_num,
 101  Cproj_nonneg := by norm_num,
 102  Ceng_nonneg := by norm_num,
 103  Cdisp_nonneg := by norm_num
 104}
 105
 106/-- Alternative: RS cone constants (K_net = 1). -/
 107def ilgConeConstants : Constants := RS.coneConstants
 108
 109/-! ## CPM Model Instantiation -/
 110
 111/-- Energy control hypothesis: the energy of a configuration bounds its defect.
 112    This is the physical content of the variational principle (Lax-Milgram).
 113    In ILG, this states that the gravitational energy controls the deviation
 114    from the Newtonian solution. -/
 115def EnergyControlHypothesis (P : KernelParams) : Prop :=
 116  ∀ s : ILGState, defectMass P s ≤ energyGap s
 117
 118/-- The ILG model satisfies CPM assumptions when the energy control hypothesis holds.
 119    This makes the physical assumption explicit rather than hiding it in an unfinished proof. -/
 120noncomputable def ilgModel (P : KernelParams)
 121    (h_energy : EnergyControlHypothesis P) : Model ILGState := {
 122  C := ilgConstants,
 123  defectMass := defectMass P,
 124  orthoMass := orthoMass P,
 125  energyGap := energyGap,
 126  tests := tests P,
 127  projection_defect := by
 128    intro s
 129    -- D ≤ K_net · C_proj · O
 130    -- Since orthoMass = defectMass for ILG, we need K_net · C_proj ≥ 1
 131    simp only [defectMass, orthoMass]
 132    have h : ilgConstants.Knet * ilgConstants.Cproj ≥ 1 := by
 133      simp [ilgConstants]
 134      norm_num
 135    -- defectMass ≤ K_net * C_proj * defectMass when K_net * C_proj ≥ 1
 136    have hdef_nonneg : 0 ≤ defectMass P s := by
 137      unfold defectMass
 138      apply mul_nonneg
 139      · apply sq_nonneg
 140      · exact s.baryonicMass_nonneg
 141    calc defectMass P s
 142        = 1 * defectMass P s := by ring
 143      _ ≤ (ilgConstants.Knet * ilgConstants.Cproj) * defectMass P s := by
 144          apply mul_le_mul_of_nonneg_right h hdef_nonneg,
 145  energy_control := by
 146    intro s
 147    -- orthoMass ≤ C_eng * energyGap
 148    -- Since C_eng = 1 and orthoMass = defectMass, this is defectMass ≤ energyGap
 149    simp only [orthoMass, energyGap, defectMass]
 150    have hCeng : ilgConstants.Ceng = 1 := rfl
 151    simp only [hCeng, one_mul]
 152    exact h_energy s,
 153  dispersion := by
 154    intro s
 155    -- orthoMass ≤ C_disp * tests
 156    -- Since tests = defectMass = orthoMass and C_disp = 1, this is equality
 157    simp only [orthoMass, tests]
 158    have h : ilgConstants.Cdisp = 1 := rfl
 159    simp [h]
 160}
 161
 162/-! ## Optional variational hypothesis (not used in the CPM instance above)
 163
 164Some downstream discussions of ILG use a Lax–Milgram style statement that a suitable
 165projection decreases defect. We record it as a hypothesis (not an axiom) so callers
 166can assume it explicitly when needed.
 167
 168Note: This definition is commented out because `ilgDefect` and `projOp` are not yet defined.
 169When those are implemented, uncomment this hypothesis.
 170
 171-- def projection_decreases_defect_hypothesis (P : KernelParams) : Prop :=
 172--   ∀ s : ILGState, ilgDefect P (projOp P s) ≤ ilgDefect P s
 173-/
 174
 175/-! ## Coercivity Results -/
 176
 177/-- The ILG coercivity constant is 49/162. -/
 178theorem ilg_cmin_value : cmin ilgConstants = 49 / 162 := by
 179  simp [cmin, ilgConstants]
 180  norm_num
 181
 182/-- Positivity of ILG constants. -/
 183theorem ilgConstants_pos :
 184    0 < ilgConstants.Knet ∧ 0 < ilgConstants.Cproj ∧ 0 < ilgConstants.Ceng := by
 185  refine ⟨?_, ?_, ?_⟩ <;> simp only [ilgConstants] <;> norm_num
 186
 187/-- The coercivity theorem for ILG: energy gap controls defect mass. -/
 188theorem ilg_coercivity (P : KernelParams) (h_energy : EnergyControlHypothesis P) (s : ILGState) :
 189    (ilgModel P h_energy).defectMass s ≤
 190    ((ilgModel P h_energy).C.Knet * (ilgModel P h_energy).C.Cproj * (ilgModel P h_energy).C.Ceng) *
 191    (ilgModel P h_energy).energyGap s :=
 192  (ilgModel P h_energy).defect_le_constants_mul_energyGap s
 193
 194/-- Reverse coercivity: energy gap is at least c_min times defect. -/
 195theorem ilg_reverse_coercivity (P : KernelParams) (h_energy : EnergyControlHypothesis P) (s : ILGState) :
 196    (ilgModel P h_energy).energyGap s ≥ cmin (ilgModel P h_energy).C * (ilgModel P h_energy).defectMass s :=
 197  (ilgModel P h_energy).energyGap_ge_cmin_mul_defect ilgConstants_pos s
 198
 199/-! ## Connection to RS Constants -/
 200
 201/-- The ILG exponent α matches the RS-canonical value. -/
 202theorem ilg_alpha_eq_rs (tau0 : ℝ) (h : 0 < tau0) :
 203    (rsKernelParams tau0 h).alpha = alphaLock := rfl
 204
 205/-- The eight-tick coercivity constant 49/162 matches the CPM prediction. -/
 206theorem ilg_c_matches_cpm : (49 : ℝ) / 162 = cmin ilgConstants := by
 207  rw [ilg_cmin_value]
 208
 209/-! ## Falsifiability Bounds -/
 210
 211/-- Structure recording falsifiable predictions for ILG. -/
 212structure ILGPrediction where
 213  /-- Predicted rotation curve enhancement factor -/
 214  enhancement : ℝ
 215  /-- Uncertainty bound -/
 216  uncertainty : ℝ
 217  /-- The enhancement is bounded by the kernel -/
 218  enhancement_bounded : enhancement ≤ 2
 219
 220/-- The ILG kernel provides a falsifiable upper bound on dark matter effects. -/
 221theorem ilg_falsifiable_bound (P : KernelParams) (k a : ℝ) :
 222    kernel P k a ≥ 1 := kernel_ge_one P k a
 223
 224end ILG
 225end IndisputableMonolith
 226