Foundation Hamiltonian Emergence Total Jcost Approx Quadratic

The quadratic approximation

In the Recognition Science framework, a ledger, a discrete record of events, assigns a cost to each possible state. States near equilibrium are described by small deviations, written as 1 + ε, where ε is a small number. The framework's cost function, J, measures the price of a deviation. The declaration totalJcost_approx_quadratic establishes a precise, machine-checked bound on how close the total cost of a multi-part state is to a simple quadratic form.

The theorem states that for any state with small deviations, the absolute difference between the total cost and the sum of the squares of the deviations (divided by 2) is at most twice the sum of the cubes of the absolute deviations. In symbols: |totalJcost(s) - quadraticEnergy(s)| ≤ 2 * Σ |ε_i|³. This is a quantitative version of the statement that J(1+ε) = ε²/2 + O(ε³). The quadratic form, the sum of ε_i²/2, is identified as the leading-order energy of the state.

This result is a proved theorem, not a hypothesis. It is derived from the fundamental scalar expansion of the cost function, which is itself proved. The bound is meaningful: for very small deviations, the cubic term becomes negligible, and the total cost is almost exactly the quadratic energy. This is the mathematical foundation for the framework's claim that a quantum Hamiltonian emerges from recognition dynamics in the small-deviation limit.

In Recognition Science, the framework models the evolution of a ledger state as a discrete process. The quadratic approximation is the link that allows this discrete process to be compared with continuous quantum evolution, specifically the Schrödinger equation. The theorem proves the scalar foundation: the cost behaves like an energy. However, the full operator-level emergence, where the recognition operator generates a self-adjoint Hamiltonian via Stone's theorem, remains a hypothesis. The framework defines the necessary structures, such as the Hilbert space embedding and the discrete evolution operator, but the proof of the operator-level statement is not yet complete.

What this means in plain terms is that the framework has rigorously shown that near equilibrium, the cost of a state is essentially a sum of independent harmonic oscillator energies. This is a significant step, as it provides a concrete, provable link between the abstract concept of recognition cost and the familiar physics of quadratic potentials. The next step, proving that this leads to a full quantum mechanical Hamiltonian, is a stated target, not an accomplished fact.

/-- 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 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 full operator-level emergence of a Hamiltonian from the recognition operator is not proved; it is a hypothesis. The theorem does not claim that the recognition dynamics are exactly equivalent to Schrödinger evolution. The result does not identify the specific physical system or interaction that the Hamiltonian describes.

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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