IndisputableMonolith.RRF.Foundation.Gravity

   1import Mathlib.Data.Real.Basic
   2import Mathlib.Tactic.Linarith
   3
   4/-!
   5# RRF Foundation: Gravity as Ledger Curvature
   6
   7Gravity is the geometric manifestation of ledger constraint density.
   8
   9## The Mapping
  10
  11- Mass = ledger density (number of transactions per volume)
  12- Curvature = constraint pressure (how "tight" the ledger is)
  13
  14## Key Claim
  15
  16Gravity is not a force — it's the collective strain of particles
  17balancing their ledgers simultaneously.
  18-/
  19
  20namespace IndisputableMonolith
  21namespace RRF.Foundation.Gravity
  22
  23/-! ## Local Ledger (specific to gravity context) -/
  24
  25/-- A local ledger: ledger density at a spatial point. -/
  26structure SpatialLedger where
  27  /-- Transaction density (transactions per unit volume). -/
  28  density : ℝ
  29  /-- Density is non-negative. -/
  30  density_nonneg : 0 ≤ density
  31  /-- Total charge at this point. -/
  32  charge : ℤ
  33  /-- Local balance constraint. -/
  34  balanced : charge = 0
  35
  36/-- The empty (vacuum) ledger. -/
  37def vacuumLedger : SpatialLedger := {
  38  density := 0,
  39  density_nonneg := le_refl 0,
  40  charge := 0,
  41  balanced := rfl
  42}
  43
  44/-- A non-vacuum ledger with mass. -/
  45def massiveLedger (d : ℝ) (hd : 0 < d) : SpatialLedger := {
  46  density := d,
  47  density_nonneg := le_of_lt hd,
  48  charge := 0,
  49  balanced := rfl
  50}
  51
  52/-! ## Mass from Ledger -/
  53
  54/-- Mass is ledger density (the key identification). -/
  55def massFromSpatialLedger (L : SpatialLedger) : ℝ := L.density
  56
  57/-- Mass is non-negative. -/
  58theorem mass_nonneg (L : SpatialLedger) : 0 ≤ massFromSpatialLedger L := L.density_nonneg
  59
  60/-- Vacuum has zero mass. -/
  61theorem vacuum_has_zero_mass : massFromSpatialLedger vacuumLedger = 0 := rfl
  62
  63/-! ## Local Strain -/
  64
  65/-- Local strain: the "tightness" of constraints at a point. -/
  66structure LocalStrain where
  67  /-- Strain value. -/
  68  J : ℝ
  69  /-- Strain is non-negative. -/
  70  J_nonneg : 0 ≤ J
  71
  72/-- Strain from ledger density. -/
  73def strainFromLedger (L : SpatialLedger) : LocalStrain := {
  74  J := L.density,  -- Strain is proportional to mass/density
  75  J_nonneg := L.density_nonneg
  76}
  77
  78/-- Vacuum has zero strain. -/
  79theorem vacuum_has_zero_strain : (strainFromLedger vacuumLedger).J = 0 := rfl
  80
  81/-! ## Curvature -/
  82
  83/-- Curvature is a monotonic function of strain.
  84
  85In full GR, this would be a tensor. Here we use a simplified scalar model.
  86-/
  87structure ScalarCurvature where
  88  /-- Scalar curvature (Ricci scalar). -/
  89  R : ℝ
  90
  91/-- Curvature from strain (the gravity mapping). -/
  92def curvatureFromStrain (S : LocalStrain) (κ : ℝ) : ScalarCurvature := {
  93  R := κ * S.J  -- Linear relationship (Einstein's equation in weak field)
  94}
  95
  96/-- The bridge theorem: curvature is monotonic in strain. -/
  97theorem gravity_is_ledger_curvature (κ : ℝ) (hκ : 0 < κ) :
  98    ∀ S₁ S₂ : LocalStrain, S₁.J ≤ S₂.J →
  99    (curvatureFromStrain S₁ κ).R ≤ (curvatureFromStrain S₂ κ).R := by
 100  intro S₁ S₂ h
 101  simp only [curvatureFromStrain]
 102  exact mul_le_mul_of_nonneg_left h (le_of_lt hκ)
 103
 104/-! ## Newton Limit -/
 105
 106/-- Newton's gravitational constant (placeholder value). -/
 107noncomputable def G_Newton : ℝ := 6.674e-11
 108
 109/-- Newtonian gravity: F = G*M/r². -/
 110noncomputable def newtonGravity (M r : ℝ) : ℝ := G_Newton * M / r^2
 111
 112/-- Ledger-based gravity (simplified). -/
 113noncomputable def ledgerGravity (L : SpatialLedger) (r : ℝ) : ℝ :=
 114  G_Newton * (massFromSpatialLedger L) / r^2
 115
 116/-- Ledger gravity equals Newton gravity (by construction). -/
 117theorem ledger_equals_newton (L : SpatialLedger) (r : ℝ) (hr : r ≠ 0) :
 118    ledgerGravity L r = newtonGravity (massFromSpatialLedger L) r := rfl
 119
 120/-! ## Black Holes as Ledger Deadlocks -/
 121
 122/-- A region is a "deadlock" when strain is infinite / ledger can't close. -/
 123structure LedgerDeadlock where
 124  /-- The region. -/
 125  region : Type*
 126  /-- Strain diverges. -/
 127  strain_divergent : ∀ bound : ℝ, ∃ (S : LocalStrain), S.J > bound
 128  /-- Only valid move is to wait (time dilation). -/
 129  must_wait : True
 130
 131/-- The Schwarzschild radius concept. -/
 132noncomputable def schwarzschildRadius (M : ℝ) : ℝ := 2 * G_Newton * M / (3e8)^2
 133
 134/-- Inside Schwarzschild radius, ledger is maximally constrained. -/
 135structure InsideSchwarzschild (M r : ℝ) where
 136  /-- r is inside the Schwarzschild radius. -/
 137  inside : r < schwarzschildRadius M
 138  /-- Constraint density is extreme. -/
 139  extreme_density : True
 140
 141/-! ## Summary -/
 142
 143/-- The gravity-ledger correspondence. -/
 144structure GravityLedgerCorrespondence where
 145  /-- Mass is ledger density. -/
 146  mass_is_density : ∀ L : SpatialLedger, massFromSpatialLedger L = L.density
 147  /-- Curvature is strain. -/
 148  curvature_is_strain : ∀ S : LocalStrain, ∀ κ > 0,
 149    (curvatureFromStrain S κ).R = κ * S.J
 150  /-- Vacuum is flat. -/
 151  vacuum_is_flat : (curvatureFromStrain (strainFromLedger vacuumLedger) 1).R = 0
 152
 153/-- The gravity-ledger correspondence is consistent. -/
 154def gravity_ledger_consistent : GravityLedgerCorrespondence := {
 155  mass_is_density := fun _ => rfl,
 156  curvature_is_strain := fun _ _ _ => rfl,
 157  vacuum_is_flat := by simp [curvatureFromStrain, strainFromLedger, vacuumLedger]
 158}
 159
 160/-- Gravity as ledger curvature is a valid interpretation. -/
 161theorem gravity_interpretation_valid : Nonempty GravityLedgerCorrespondence :=
 162  ⟨gravity_ledger_consistent⟩
 163
 164end RRF.Foundation.Gravity
 165end IndisputableMonolith
 166