132def diagonalHamiltonian (N : ℕ) : DiscreteEvolution N where 133 hamiltonian := fun i j => if i = j then 1 else 0
proof body
Definition body.
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) -/
used by (1)
From the project-wide theorem graph. These declarations reference this one in their body.