DiscreteEvolution

formal source

 119
 120/-- The discrete evolution operator at small strain: applies R̂ in the
 121    quadratic regime, parameterized by a real "Hamiltonian matrix" H. -/
 122structure DiscreteEvolution (N : ℕ) where
 123  hamiltonian : Fin N → Fin N → ℝ
 124  symmetric : ∀ i j, hamiltonian i j = hamiltonian j i
 125
 126/-- Apply one step of discrete evolution to deviations (linearized). -/
 127def DiscreteEvolution.step (ev : DiscreteEvolution N) (ψ : DeviationHilbert N) :
 128    DeviationHilbert N :=
 129  fun i => ψ i - Complex.I * (Finset.univ.sum fun j => (ev.hamiltonian i j : ℂ) * ψ j)
 130
 131/-- The diagonal Hamiltonian: H_ii = 1 (from J''(1) = 1 calibration). -/
 132def diagonalHamiltonian (N : ℕ) : DiscreteEvolution N where
 133  hamiltonian := fun i j => if i = j then 1 else 0
 134  symmetric := by intro i j; by_cases h : i = j <;> simp [h, eq_comm]
 135
 136/-! ## Emergence Hypothesis -/
 137
 138/-- **HYPOTHESIS**: The Recognition Operator generates a self-adjoint
 139    Hamiltonian in the small-deviation limit.
 140
 141    STATUS: HYPOTHESIS — the scalar foundation is proved (quadratic
 142    emergence + remainder bounds). The operator-level statement requires:
 143    1. Stone's theorem for discrete unitary groups (not in Mathlib)
 144    2. A proof that R̂ evolution on LedgerState near equilibrium is
 145       approximated by the linear step defined above
 146
 147    PROOF ROADMAP:
 148    - Define U_Δ(ψ) = embed(R̂(unembed(ψ))) for small ψ
 149    - Show U_Δ is approximately unitary: ‖U_Δ ψ‖² = ‖ψ‖² + O(ε³)
 150    - Apply discrete Stone: generator of {U_Δ^n} is self-adjoint
 151    - Identify generator with diagonalHamiltonian (from J''(1) = 1) -/
 152def H_HamiltonianIsGenerator (N : ℕ) : Prop :=