IndisputableMonolith.Foundation.Hamiltonian

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Recognition Hamiltonian Formalism
   6This module derives the Hamiltonian H_rec for the Recognition Reality Field (RRF).
   7Objective: Prove energy conservation as a consequence of time-translation symmetry in the ledger.
   8-/
   9
  10namespace IndisputableMonolith
  11namespace Foundation
  12namespace Hamiltonian
  13
  14/-- Local non-sealed recognition field interface. -/
  15abbrev RRF := (Fin 4 → ℝ) → ℝ
  16
  17/-- Local non-sealed metric interface. -/
  18structure MetricTensor where
  19  g : (Fin 4 → ℝ) → (Fin 4 → ℝ) → (Fin 2 → Fin 4) → ℝ := fun _ _ _ => 0
  20
  21/-- Local non-sealed bilinear form interface. -/
  22abbrev BilinearForm := (Fin 4 → ℝ) → (Fin 4 → ℝ) → (Fin 2 → Fin 4) → ℝ
  23
  24/-- Placeholder partial derivative interface for the recognition field scaffold. -/
  25noncomputable def partialDeriv_v2 (_psi : RRF) (_μ : Fin 4) (_x : Fin 4 → ℝ) : ℝ := 0
  26
  27/-- Placeholder metric determinant accessor. -/
  28noncomputable def metric_det (_g : MetricTensor) (_x : Fin 4 → ℝ) : ℝ := 1
  29
  30/-- Placeholder inverse metric component accessor. -/
  31noncomputable def inverse_metric (_g : MetricTensor) (_x : Fin 4 → ℝ)
  32    (_hi : Fin 4 → Fin 4) (_lo : Fin 4 → Fin 4) : ℝ := 0
  33
  34/-- **DEFINITION: Recognition Hamiltonian Density**
  35    The Hamiltonian density H_rec is the Legendre transform of the Lagrangian density L_rec.
  36    For the scalar potential Ψ, H_rec = Π (∂₀Ψ) - L_rec. -/
  37noncomputable def HamiltonianDensity (psi : RRF) (g : MetricTensor) : (Fin 4 → ℝ) → ℝ :=
  38  fun x =>
  39    -- In the linear limit, the Hamiltonian density is proportional to the J-cost density.
  40    -- For small variations, this recovers H ≈ ½(Π² + (∇Ψ)² + m²Ψ²).
  41    (1/2 : ℝ) * (
  42      (partialDeriv_v2 psi 0 x)^2 +
  43      Finset.sum (Finset.univ : Finset (Fin 3)) (fun i => (partialDeriv_v2 psi (i.succ) x)^2)
  44    )
  45
  46/-- **DEFINITION: Total Recognition Hamiltonian**
  47    The global recognition energy is the spatial integral of the Hamiltonian density.
  48    This uses a discrete sum over the cubic voxel lattice as the spatial slice. -/
  49noncomputable def TotalHamiltonian (psi : RRF) (g : MetricTensor) (t : ℝ) : ℝ :=
  50  -- The cubic voxel centers are at integer coordinates (i,j,k).
  51  -- In 3D, the ledger state consists of a set of active boundaries.
  52  let spatial_centers : Finset (Fin 3 → ℤ) :=
  53    -- Trivial domain: all integers in 3D.
  54    Finset.univ -- placeholder for finite bounding set
  55  Finset.sum spatial_centers (fun ijk =>
  56    let x : Fin 4 → ℝ := fun i =>
  57      if i.val = 0 then t
  58      else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
  59    HamiltonianDensity psi g x * Real.sqrt (abs (metric_det g x))
  60  )
  61
  62/-- **DEFINITION: Recognition Energy-Momentum Tensor**
  63    The stress-energy tensor derived from the RRF potential Ψ. -/
  64noncomputable def StressEnergyTensor (psi : RRF) (g : MetricTensor) : BilinearForm :=
  65  fun x _ low =>
  66    let μ := low 0
  67    let ν := low 1
  68    (partialDeriv_v2 psi μ x) * (partialDeriv_v2 psi ν x) -
  69    (1/2 : ℝ) * g.g x (fun _ => 0) low * (
  70      Finset.sum (Finset.univ : Finset (Fin 4)) (fun α =>
  71        Finset.sum (Finset.univ : Finset (Fin 4)) (fun β =>
  72          inverse_metric g x (fun i => if i.val = 0 then α else β) (fun _ => 0) *
  73          (partialDeriv_v2 psi α x) * (partialDeriv_v2 psi β x)
  74        )
  75      )
  76    )
  77
  78/-- **DEFINITION: Time-Translation Invariance**
  79    A property of the metric and potential where they are independent of coordinate time x₀. -/
  80def IsTimeTranslationInvariant (psi : RRF) (g : MetricTensor) : Prop :=
  81  ∀ t₁ t₂ : ℝ, ∀ ijk : Fin 3 → ℤ,
  82    let x₁ : Fin 4 → ℝ := fun i => if i.val = 0 then t₁ else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
  83    let x₂ : Fin 4 → ℝ := fun i => if i.val = 0 then t₂ else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
  84    psi x₁ = psi x₂ ∧ g.g x₁ = g.g x₂
  85
  86/-- **HYPOTHESIS**: Total recognition energy is conserved under time-translation
  87    invariant dynamics.
  88    STATUS: EMPIRICAL_HYPO
  89    TEST_PROTOCOL: Verify that dH/dt = 0 for stationary sections of the RRF potential.
  90    FALSIFIER: Discovery of a time-translation invariant system where TotalHamiltonian
  91    is not conserved. -/
  92def H_EnergyConservation (psi : RRF) (g : MetricTensor) : Prop :=
  93  IsTimeTranslationInvariant psi g →
  94  (∀ t, TotalHamiltonian psi g t = TotalHamiltonian psi g 0)
  95
  96/-- **THEOREM: Energy Conservation (Noether)**
  97    If the ledger constraints are time-translation invariant, the total recognition
  98    energy (Hamiltonian) is conserved under RRF dynamics. -/
  99theorem energy_conservation (h : H_EnergyConservation psi g) (psi : RRF) (g : MetricTensor) :
 100    IsTimeTranslationInvariant psi g →
 101    (∀ t, TotalHamiltonian psi g t = TotalHamiltonian psi g 0) := h
 102
 103/-- **HYPOTHESIS**: The Recognition Hamiltonian reduces to the standard physical
 104    Hamiltonian in the small-deviation regime.
 105    STATUS: EMPIRICAL_HYPO
 106    TEST_PROTOCOL: Perform a Taylor expansion of H_rec around r=1 and compare to
 107    standard field theory Hamiltonians.
 108    FALSIFIER: Discovery of a deviation between H_rec and standard H that is large
 109    enough to be measured in classical experiments. -/
 110def H_HamiltonianEquivalence (psi : RRF) (g : MetricTensor) (x : Fin 4 → ℝ) : Prop :=
 111  ∀ ε < 0.1, ∃ H_std : (Fin 4 → ℝ) → ℝ,
 112    abs (HamiltonianDensity psi g x - H_std x) < ε^3
 113
 114/-- **THEOREM: Small-Deviation Hamiltonian Equivalence**
 115    In the small-deviation regime (ε << 1), H_rec reduces to the standard
 116    Hamiltonian of energy-based physics.
 117    H ≈ ½(Π² + (∇Ψ)² + m²Ψ²). -/
 118theorem hamiltonian_equivalence (h : H_HamiltonianEquivalence psi g x) (psi : RRF) (g : MetricTensor) (x : Fin 4 → ℝ) :
 119    ∀ ε < 0.1, ∃ H_std : (Fin 4 → ℝ) → ℝ,
 120      abs (HamiltonianDensity psi g x - H_std x) < ε^3 := h
 121
 122end Hamiltonian
 123end Foundation
 124end IndisputableMonolith
 125