Foundation Pair Kernel Coherence Scaled Event Operator

The coherence-scaled event operator

The foundation pair kernel coherence scaled event operator is a mathematical object in Recognition Science: a discrete quantum evolution operator built from one primitive posting, with no free real coefficients. It is defined on a finite state space (a torus of size 3 per dimension) and carries a single native energy scale, the event generator energy scale, which equals the framework constant E_coh. The operator's evolution is generated by a Hamiltonian that is Hermitian and satisfies Stone's theorem, meaning it produces a valid unitary time evolution.

The construction starts with a realized primitive posting, which is a pair of events that the framework's ledger records as actually occurring. From that posting, the construction derives a potential field as the negative of the signed real Green's function for a unit posting, and then forms the one-body Hamiltonian using that potential and the common scale. The source response is defined as the real-linear covector obtained by varying the scalar potential energy, and the Born potential energy is the sum over sites of the potential times the squared complex norm of the wavefunction.

The construction proves several structural theorems: the operator is Hermitian, it satisfies Stone's evolution condition, the source response is linear in the potential variation, and the dimensionless coupling ratio is fixed to one. It also proves that doubling the source strength changes the operator, so the construction is not degenerate. The whole operator is certified as a valid recognition one-body quantum attachment, meaning it fits the framework's interface for a quantum system derived from recognition events.

In Recognition Science, this construction is a model carrier with theorem properties. The choice of the shared unit-posting operator is new physical structure, not forced by prior premises. The construction does not prove that this operator is the molecular Hamiltonian, does not prove source-action duality, and does not prove the physical correctness of the emitted spectrum. It is an executable object that a deterministic program can now put in front of atomic reality, as a test of whether the framework's structure matches measurement.

theorem coherenceScaledEventOperator_isHermitian
    (op : CoherenceScaledEventOperator3) :
    (Hc op.evolution).IsHermitian :=
  oneBodyHamiltonian3_isHermitian _ _ _

theorem coherenceScaledEventOperator_stone
    (op : CoherenceScaledEventOperator3) :
    StoneGeneratorCert (TorusCard3 3) op.evolution :=
  oneBodyHamiltonian3_stone _ _ _

/-- This MODEL emits dimensionless source/kinetic ratio one. The theorem is
an audit of the constructor, not a physical-selection theorem. -/
theorem dimensionless_couplingRatio_eq_one :
    oneBodyCouplingRatio
        eventGeneratorEnergyScale eventGeneratorEnergyScale =
      1 := by
  unfold oneBodyCouplingRatio
  exact div_self (ne_of_gt eventGeneratorEnergyScale_pos)

/-- Exact scaled line response. The source-to-density response and the
Hamiltonian generator carry the same single native event scale. -/
theorem coherenceBornPotentialEnergy3_line
    (potential variation : PotentialVariation3)
    (ψ : OneBodyHilbert3) (t : ℝ) :
    eventGeneratorEnergyScale *
        bornPotentialEnergy3
          (fun i => potential i + t * variation i) ψ =
      eventGeneratorEnergyScale *
          bornPotentialEnergy3 potential ψ +
        t * coherenceBornSourceResponse3 ψ variation := by
  rw [bornPotentialEnergy3_line]
  simp [coherenceBornSourceResponse3]
  ring

/-- The carrier is a concrete instance of the old attachment predicate. This
states compatibility only; that predicate does not force physical identity. -/
theorem coherenceScaledEventOperator_witnesses_model_attachment
    (op : CoherenceScaledEventOperator3) :
    RecognitionOneBodyQuantumAttachment3 op.evolution := by
  refine
    ⟨op.event, op.event_realized,
      eventGeneratorEnergyScale, eventGeneratorEnergyScale,
      1, 0, eventGeneratorEnergyScale_pos,
      eventGeneratorEnergyScale_pos, by norm_num, ?_⟩
  rw [CoherenceScaledEventOperator3.evolution,
    unitPostingPotential3_eq_q1_model]

What this page does not claim

This construction does not prove that the operator is the molecular Hamiltonian. This construction does not prove source-action duality. This construction does not prove the physical correctness of the emitted spectrum.

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

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