Foundation Hamiltonian Emergence Embed Norm Sq

The embedding's norm

In quantum mechanics, the squared length of a state vector in Hilbert space is its probability or its energy. The declaration embed_norm_sq establishes a precise relationship between that squared length and a classical energy expression, within the Recognition Science framework. The framework models physical states as a ledger, a discrete record of events, and measures the cost of deviations from equilibrium. Near equilibrium, each bond multiplier is written as 1 + εᵢ, where εᵢ is a small real number.

The theorem states that the sum of the squared norms of the embedded deviations equals the sum of the squares of the deviations themselves. In symbols, Σ |embed(s)(i)|² = Σ (s.deviations(i))². This is not an approximation; it is an exact identity, proved in the machine-checked library of formal theorems. The embedding maps real deviations into a complex Hilbert space, and the theorem shows that the squared norm of that embedding is exactly twice the quadratic energy, since the quadratic energy is defined as half the sum of squares. This is the first concrete link between the ledger's geometry and the energy of a quantum state.

The scalar foundation is proved: the cost function J(1+ε) expands as ε²/2 + c·ε³ with |c| ≤ 2, making the quadratic form exact at leading order. The theorem embed_norm_sq builds on this to show the norm-energy identity holds for any small-deviation state. What remains unproved is the operator-level emergence: the claim that the recognition operator generates a self-adjoint Hamiltonian via Stone's theorem. That requires Hilbert space infrastructure for discrete systems not yet in the library. The hypothesis is stated, the scalar part is proved, and the operator part is a target.

/-- 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 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

/-- 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]

What this page does not claim

The operator-level emergence of a self-adjoint Hamiltonian is not proved. The embedding does not establish unitarity of the discrete evolution. The theorem does not identify the Hamiltonian matrix beyond the diagonal case.

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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Derived articles

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