Foundation Hamiltonian Emergence Per Bond Remainder Bounded

The bounded remainder

The Recognition Science framework models physical structure as the cost of recognition events, where a recognition event is a discrete act of matching one state to another. Its central cost function, J(x) = (x + 1/x)/2 - 1, assigns a price to any deviation of a bond multiplier x from its equilibrium value of 1. Near that equilibrium, when x = 1 + ε with ε small, the cost looks almost like a simple parabola: J(1 + ε) = ε²/2 + O(ε³). The theorem per_bond_remainder_bounded makes that approximation precise and quantitative.

In plain language, the theorem states that for any small deviation |ε| ≤ 1/2, the absolute difference between the true cost J(1 + ε) and its quadratic approximation ε²/2 is at most 2|ε|³. This is a bound on the cubic remainder term: it says the error shrinks at least as fast as the cube of the deviation. The proof is a direct consequence of the stronger expansion theorem quadratic_emergence, which guarantees the existence of a coefficient c with |c| ≤ 2 such that the remainder is exactly c·ε³. The bound then follows by taking absolute values and using the fact that |c| ≤ 2.

The framework's library, a machine-checked collection of formal theorems, uses this scalar result as the foundation for a larger claim about Hamiltonian emergence. The idea is that the total cost of a state with many small deviations is approximately the sum of the quadratic terms, and the error is bounded by a sum of cubic terms. This approximation is then embedded into a complex Hilbert space, where the quadratic energy becomes half the squared norm of the embedded state. The discrete evolution operator, defined as a linear step involving a symmetric Hamiltonian matrix, is intended to approximate the recognition dynamics in this small-deviation regime.

What the theorem does not claim is the operator-level emergence itself. The scalar expansion and its remainder bound are proved, but the step from this scalar foundation to a full self-adjoint Hamiltonian operator acting on Hilbert space remains a hypothesis. The library explicitly labels the operator-level statement as HYPOTHESIS, noting that it would require Stone's theorem for discrete unitary groups, which is not yet available in the underlying formal library, and a proof that the recognition operator's evolution is approximated by the linear step. The theorem emergence_scalar_proved shows that the scalar part is indeed proved, but it does not establish the existence of the Hamiltonian as a generator of the dynamics.

In the framework's own terms, the bounded remainder is the precise sense in which the cost function is quadratic near equilibrium. It is the quantitative control that makes the quadratic approximation trustworthy for small deviations, and it is the proved scalar anchor on which the unproved operator-level emergence hypothesis rests. The practical consequence is that for small deviations, the framework's recognition cost behaves like a harmonic oscillator energy, with the cubic correction bounded and controlled.

/-- 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]

/-- 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]

/-- **HYPOTHESIS**: The Recognition Operator generates a self-adjoint
    Hamiltonian in the small-deviation limit.

    STATUS: HYPOTHESIS — the scalar foundation is proved (quadratic
    emergence + remainder bounds). The operator-level statement requires:
    1. Stone's theorem for discrete unitary groups (not in Mathlib)
    2. A proof that R̂ evolution on LedgerState near equilibrium is
       approximated by the linear step defined above

    PROOF ROADMAP:
    - Define U_Δ(ψ) = embed(R̂(unembed(ψ))) for small ψ
    - Show U_Δ is approximately unitary: ‖U_Δ ψ‖² = ‖ψ‖² + O(ε³)
    - Apply discrete Stone: generator of {U_Δ^n} is self-adjoint
    - Identify generator with diagonalHamiltonian (from J''(1) = 1) -/
def H_HamiltonianIsGenerator (N : ℕ) : Prop :=
  ∃ (ev : DiscreteEvolution N),
    ∀ (s : SmallDeviationState N),
      |totalJcost s - quadraticEnergy s| ≤
        2 * Finset.univ.sum fun i => |s.deviations i| ^ 3

What this page does not claim

The theorem does not prove that the recognition operator generates a self-adjoint Hamiltonian. The theorem does not establish the unitary approximation of the recognition dynamics by the linear step. The theorem does not claim that the cubic bound is tight; it only provides an upper bound.

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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