RRF

formal source

  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