H_EnergyConservation

formal source

  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