IndisputableMonolith.Relativity.ILG.Action

   1import Mathlib
   2import IndisputableMonolith.Relativity.Geometry
   3import IndisputableMonolith.Relativity.Fields
   4
   5namespace IndisputableMonolith
   6namespace Relativity
   7namespace ILG
   8
   9/-- Re-export geometry and field types for ILG use. -/
  10abbrev Metric := Geometry.MetricTensor
  11abbrev RefreshField := Fields.ScalarField
  12abbrev VolumeElement := Fields.VolumeElement
  13
  14/-- Einstein–Hilbert action: S_EH = (M_P²/2) ∫ √(-g) R d^4x.
  15    Scaffold: returns symbolic R evaluation (integration machinery pending). -/
  16noncomputable def EHAction (g : Metric) : ℝ :=
  17  -- Placeholder integration over spacetime volume
  18  -- Should be: integral over manifold of √(-g(x)) * R(x)
  19  let x₀ : Fin 4 → ℝ := fun _ => 0  -- Sample point
  20  Geometry.ricci_scalar g x₀  -- Scaffold: use single-point value
  21
  22/-- Alias for consistency. -/
  23noncomputable def S_EH := EHAction
  24
  25/-- Default volume element for action integrals. -/
  26noncomputable def default_volume : VolumeElement :=
  27  { grid_spacing := 1.0, grid_spacing_pos := by norm_num }
  28
  29/-- ψ-sector kinetic term: (α/2) ∫ √(-g) g^{μν} (∂_μ ψ)(∂_ν ψ) d^4x.
  30    Now uses actual Fields.kinetic_action. -/
  31noncomputable def PsiKinetic (g : Metric) (ψ : RefreshField) (α : ℝ) : ℝ :=
  32  α * Fields.kinetic_action ψ g default_volume
  33
  34/-- ψ-sector mass/potential term: (C_lag/2) ∫ √(-g) ψ² d^4x.
  35    Now uses actual Fields.potential_action. -/
  36noncomputable def PsiPotential (g : Metric) (ψ : RefreshField) (C_lag : ℝ) : ℝ :=
  37  Fields.potential_action ψ (C_lag ^ 2) g default_volume
  38
  39/-- ψ-sector action placeholder parameterised by (C_lag, α): kinetic + potential. -/
  40noncomputable def PsiAction (g : Metric) (ψ : RefreshField) (C_lag α : ℝ) : ℝ :=
  41  PsiKinetic g ψ α + PsiPotential g ψ C_lag
  42
  43/-- Global parameter bundle for ILG (α, C_lag). -/
  44structure ILGParams where
  45  alpha : ℝ
  46  cLag  : ℝ
  47  deriving Inhabited
  48
  49/-- Index conventions (symbolic): use natural numbers as abstract tensor indices. -/
  50abbrev Index : Type := Nat
  51
  52/-- Kronecker delta δᵤᵥ (symbolic). -/
  53@[simp] noncomputable def kron (μ ν : Index) : ℝ := if μ = ν then 1 else 0
  54
  55/-- Raise/lower index placeholders (identity maps in the scaffold). -/
  56@[simp] def raiseIndex (μ : Index) : Index := μ
  57@[simp] def lowerIndex (μ : Index) : Index := μ
  58
  59/-- Variation notation scaffolding: delta of a scalar expression (symbolic identity). -/
  60@[simp] noncomputable def deltaVar (x : ℝ) : ℝ := x
  61
  62/-- Functional derivative: δS/δx for action functional S and state variable x. -/
  63noncomputable def dS_dx (S_func : ℝ → ℝ) (x₀ : ℝ) : ℝ :=
  64  deriv S_func x₀
  65
  66
  67/-- Symbolic ILG Lagrangian density (toy): L = (∂ψ)^2/2 − m^2 ψ^2/2 + cLag·alpha.
  68    Here we treat all terms as scalars to keep the scaffold compiling. -/
  69noncomputable def L_density (_g : Metric) (_ψ : RefreshField) (p : ILGParams) : ℝ :=
  70  (p.alpha ^ 2) / 2 - (p.cLag ^ 2) / 2 + p.cLag * p.alpha
  71
  72/-- Covariant scalar Lagrangian pieces (symbolic). -/
  73noncomputable def L_kin (_g : Metric) (_ψ : RefreshField) (p : ILGParams) : ℝ := (p.alpha ^ 2) / 2
  74noncomputable def L_mass (_g : Metric) (_ψ : RefreshField) (p : ILGParams) : ℝ := (p.cLag ^ 2) / 2
  75/-- Potential Lagrangian - depends on metric and refresh field configuration.
  76    Placeholder using coupling constant scaled by field variance. -/
  77noncomputable def L_pot (_g : Metric) (_ψ : RefreshField) (p : ILGParams) : ℝ := p.cLag / 2
  78noncomputable def L_coupling (_g : Metric) (_ψ : RefreshField) (p : ILGParams) : ℝ := p.cLag * p.alpha
  79
  80/-- Covariant scalar Lagrangian (toy): L_cov = L_kin − L_mass + L_pot + L_coupling. -/
  81noncomputable def L_cov (g : Metric) (ψ : RefreshField) (p : ILGParams) : ℝ :=
  82  L_kin g ψ p - L_mass g ψ p + L_pot g ψ p + L_coupling g ψ p
  83
  84/-- Covariant total action using L_cov: S_cov = S_EH + ∫ L_cov (toy: scalar sum). -/
  85noncomputable def S_total_cov (g : Metric) (ψ : RefreshField) (p : ILGParams) : ℝ :=
  86  S_EH g + L_cov g ψ p
  87
  88/-- GR-limit for S_total_cov (α=0, C_lag=0). -/
  89theorem gr_limit_cov (g : Metric) (ψ : RefreshField) :
  90  S_total_cov g ψ { alpha := 0, cLag := 0 } = S_EH g := by
  91  unfold S_total_cov L_cov L_kin L_mass L_pot L_coupling
  92  simp
  93
  94/-- Convenience total action using bundled params. -/
  95noncomputable def S_total (g : Metric) (ψ : RefreshField) (p : ILGParams) : ℝ :=
  96  S_EH g + PsiAction g ψ p.cLag p.alpha
  97
  98/-- ψ-sector action using bundled parameters. -/
  99noncomputable def PsiActionP (g : Metric) (ψ : RefreshField) (p : ILGParams) : ℝ :=
 100  PsiKinetic g ψ p.alpha + PsiPotential g ψ p.cLag
 101
 102/-! Euler-Lagrange predicates moved to ILG/Variation.lean (now use real equations).
 103    EL_g and EL_psi now defined in Variation.lean with actual PDEs. -/
 104
 105/-- Consolidated bands schema for observables (scaffold). -/
 106structure Bands where
 107  κ_ppn : ℝ
 108  κ_lensing : ℝ
 109  κ_gw : ℝ
 110  h_ppn : 0 ≤ κ_ppn
 111  h_lensing : 0 ≤ κ_lensing
 112  h_gw : 0 ≤ κ_gw
 113
 114/-- Map ILG parameters to a bands schema (toy: proportional to |C_lag·α|). -/
 115noncomputable def bandsFromParams (p : ILGParams) : Bands :=
 116  let κ := |p.cLag * p.alpha|
 117  { κ_ppn := κ, κ_lensing := κ, κ_gw := κ
 118  , h_ppn := by exact abs_nonneg _
 119  , h_lensing := by exact abs_nonneg _
 120  , h_gw := by exact abs_nonneg _ }
 121
 122/-! Symbolic Einstein equations moved to Variation/Einstein.lean.
 123    VacuumEinstein now defined with real G_μν = 0. -/
 124
 125/-- Bundle the action inputs `(g, ψ)` for convenience in downstream modules. -/
 126abbrev ActionInputs := Metric × RefreshField
 127
 128/-- Apply total action on bundled inputs. -/
 129noncomputable def S_on (inp : ActionInputs) (p : ILGParams) : ℝ :=
 130  S_total inp.fst inp.snd p
 131
 132/-- Full ILG action: S[g, ψ; C_lag, α] := S_EH[g] + S_ψ[g,ψ]. -/
 133noncomputable def S (g : Metric) (ψ : RefreshField) (C_lag α : ℝ) : ℝ :=
 134  S_EH g + PsiAction g ψ C_lag α
 135
 136/-- GR-limit reduction: when C_lag = 0 and α = 0, the ψ-sector vanishes. -/
 137theorem gr_limit_reduces (g : Metric) (ψ : RefreshField) :
 138  S g ψ 0 0 = S_EH g := by
 139  unfold S PsiAction PsiKinetic PsiPotential
 140  simp [Fields.kinetic_action, Fields.potential_action]
 141
 142/-- GR-limit for bundled parameters (α=0, C_lag=0). -/
 143theorem gr_limit_zero (g : Metric) (ψ : RefreshField) :
 144  S_total g ψ { alpha := 0, cLag := 0 } = S_EH g := by
 145  unfold S_total PsiAction PsiKinetic PsiPotential
 146  simp [Fields.kinetic_action, Fields.potential_action]
 147
 148/-- GR-limit for bundled inputs. -/
 149theorem gr_limit_on (inp : ActionInputs) :
 150  S_on inp { alpha := 0, cLag := 0 } = S_EH inp.fst := by
 151  unfold S_on S_total
 152  exact gr_limit_reduces inp.fst inp.snd
 153
 154end ILG
 155end Relativity
 156end IndisputableMonolith
 157