Foundation Hamiltonian Emergence Discrete Evolution

The discrete evolution operator

In quantum mechanics, the Schrödinger equation describes how a quantum state changes over time. The Recognition Science framework's DiscreteEvolution declaration is a discrete, linearized version of that idea. It models a single step of evolution for a system of N bonds, where each bond is a connection that can deviate slightly from its equilibrium value. The step is defined by a real symmetric matrix H, called the Hamiltonian, which encodes the interactions between bonds. The evolution rule is ψ′(i) = ψ(i) − i·Σⱼ H(i,j)ψ(j), which is the standard first-order approximation to continuous Schrödinger evolution with time step Δt. This is a definitional choice, a model of how the framework's recognition dynamics might look in the small-deviation regime.

The framework provides a specific default Hamiltonian: the diagonal matrix with all entries equal to 1. This choice is not arbitrary. It comes from the second derivative of the J-cost function at unity, J″(1) = 1, which is a calibration condition in the framework. The scalar foundation for this emergence is proved in the machine-checked library: for a small deviation ε, the cost J(1+ε) equals ε²/2 plus a remainder bounded by 2|ε|³. This means that near equilibrium, the cost function behaves exactly like a quadratic form, which is the kinetic energy of the emergent Hamiltonian. The total cost of a multi-bond state is approximated by the sum of ½εᵢ², with the error bounded by a cubic term. This is the proved scalar foundation: the cost function is quadratic near unity, and the quadratic energy is the leading-order term.

The declaration also defines an embedding of real deviations into a complex Hilbert space ℂᴺ, with the standard inner product. The squared norm of this embedding equals twice the quadratic energy. This is a formal bridge between the framework's real-valued ledger states and the complex vector space where quantum mechanics lives. The DiscreteEvolution structure itself is a model: it defines what a step looks like, parameterized by a Hamiltonian matrix. The step function is a definition, not a theorem. The framework's library proves that the scalar part of the emergence hypothesis holds: there exists a DiscreteEvolution (the diagonal one) such that the total J-cost is approximated by the quadratic energy within the stated bound.

What the declaration does not claim is the full operator-level emergence. The hypothesis that the Recognition Operator generates a self-adjoint Hamiltonian via Stone's theorem remains unproved. The library states this as a hypothesis, not a theorem, because the required Hilbert space infrastructure for discrete systems is not yet in the formal library. The step function is a linearized approximation, not a proof that the full recognition dynamics are unitary. The diagonal Hamiltonian is a specific model, not a derived consequence. The framework proves the scalar foundation: the cost is quadratic near unity. The operator-level statement, that the recognition dynamics are equivalent to discrete Schrödinger evolution, is a target for future work, not an established result.

In plain terms, DiscreteEvolution is a carefully scoped tool. It provides the language to state the emergence hypothesis and proves the scalar cost expansion that would ground it. It does not, by itself, derive quantum mechanics from recognition. The step from a quadratic cost function to a full unitary evolution operator is the open problem. The declaration marks the boundary: here is the model, here is the proved scalar foundation, and here is the hypothesis that remains to be proved.

/-- The scalar J-cost expansion: J(1+ε) = ε²/2 + c·ε³ with |c| ≤ 2.
    This is the fundamental lemma: J-cost IS a quadratic form near unity. -/
theorem quadratic_emergence (ε : ℝ) (hε : |ε| ≤ 1 / 2) :
    ∃ c : ℝ, Jcost (1 + ε) = ε ^ 2 / 2 + c * ε ^ 3 ∧ |c| ≤ 2 :=
  Jcost_one_plus_eps_quadratic ε hε

/-- The cubic remainder per bond is bounded. -/
theorem per_bond_remainder_bounded (ε : ℝ) (hε : |ε| ≤ 1 / 2) :
    |Jcost (1 + ε) - ε ^ 2 / 2| ≤ 2 * |ε| ^ 3 := by
  obtain ⟨c, hc_eq, hc_bound⟩ := quadratic_emergence ε hε
  rw [hc_eq]
  have : ε ^ 2 / 2 + c * ε ^ 3 - ε ^ 2 / 2 = c * ε ^ 3 := by ring
  rw [this, abs_mul]
  calc |c| * |ε ^ 3|
      ≤ 2 * |ε ^ 3| := by nlinarith [abs_nonneg (ε ^ 3)]
    _ = 2 * |ε| ^ 3 := by rw [abs_pow]

/-- Total J-cost approximates quadratic energy for small deviations. -/
theorem totalJcost_approx_quadratic (s : SmallDeviationState N) :
    |totalJcost s - quadraticEnergy s| ≤
    2 * Finset.univ.sum fun i => |s.deviations i| ^ 3 := by
  unfold totalJcost quadraticEnergy
  calc |Finset.univ.sum (fun i => Jcost (1 + s.deviations i)) -
        Finset.univ.sum (fun i => (s.deviations i) ^ 2 / 2)|
      = |Finset.univ.sum (fun i =>
          Jcost (1 + s.deviations i) - (s.deviations i) ^ 2 / 2)| := by
        congr 1; rw [← Finset.sum_sub_distrib]
    _ ≤ Finset.univ.sum (fun i =>
          |Jcost (1 + s.deviations i) - (s.deviations i) ^ 2 / 2|) :=
        Finset.abs_sum_le_sum_abs _ _
    _ ≤ Finset.univ.sum (fun i => 2 * |s.deviations i| ^ 3) := by
        apply Finset.sum_le_sum
        intro i _
        exact per_bond_remainder_bounded (s.deviations i) (s.small i)
    _ = 2 * Finset.univ.sum (fun i => |s.deviations i| ^ 3) := by
        rw [← Finset.mul_sum]

/-- The squared norm of the embedding equals twice the quadratic energy. -/
theorem embed_norm_sq (s : SmallDeviationState N) :
    (Finset.univ.sum fun i => Complex.normSq (embed s i)) =
    Finset.univ.sum fun i => (s.deviations i) ^ 2 := by
  apply Finset.sum_congr rfl
  intro i _
  simp [embed, Complex.normSq_ofReal]
  ring

/-- The discrete evolution operator at small strain: applies R̂ in the
    quadratic regime, parameterized by a real "Hamiltonian matrix" H. -/
structure DiscreteEvolution (N : ℕ) where
  hamiltonian : Fin N → Fin N → ℝ
  symmetric : ∀ i j, hamiltonian i j = hamiltonian j i

/-- The scalar part of the emergence hypothesis is already proved. -/
theorem emergence_scalar_proved (N : ℕ) :
    H_HamiltonianIsGenerator N :=
  ⟨diagonalHamiltonian N, totalJcost_approx_quadratic⟩

What this page does not claim

The declaration does not prove that the Recognition Operator generates a self-adjoint Hamiltonian. The step function is not shown to be unitary in the full recognition dynamics. The diagonal Hamiltonian is a model choice, not a derived consequence of the framework.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/HamiltonianEmergence.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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