Foundation Pair Kernel Coherence Scaled Event Operator Coherence Scaled Event Op

The attachment witness

The declaration coherenceScaledEventOperator_witnesses_model_attachment is a theorem in the Recognition Science library's machine-checked collection of formal theorems. It states that a particular operator, CoherenceScaledEventOperator3, satisfies the framework's RecognitionOneBodyQuantumAttachment3 property. That property is the framework's formal certificate that an object can serve as a one-body quantum model: the operator carries a realized primitive posting, has a positive energy scale, and its evolution is built from that posting through the framework's standard constructions. The theorem proves this by exhibiting the required data explicitly, using the operator's own event, its realized status, and its energy scale.

The operator itself is a model, not a derived law. It is defined by choosing a single shared unit-posting potential, which fixes the dimensionless coupling ratio to one. The framework's premises do not prove that this operator is the molecular Hamiltonian, nor that its emitted spectrum is physically correct. The theorem establishes only that the operator meets the formal attachment criteria, not that those criteria select the right physics. The library's own documentation states this honestly: the module is a model carrier with theorem properties.

What the theorem does prove, as a formal result, is a set of structural properties. The operator is Hermitian, it satisfies Stone's theorem for unitary evolution, its source response is linear, and the source attachment condition holds. These are machine-checked theorems, not assumptions. The declaration also proves that the operator is not trivially insensitive to its source: doubling the source scale changes the Hamiltonian, so the model distinguishes between different source configurations. This matters because it shows the attachment certificate is not vacuous; it actually constrains the operator's behavior.

The declaration does not claim that this operator is the one realized in nature. It does not prove RemainingPhysicalEquality, source-action duality, or the physical correctness of the emitted spectrum. It does not use any hydrogen or CODATA input, and it introduces no new axioms. The theorem is a formal bridge: it shows that a specific, well-defined object fits the framework's attachment interface, and nothing more. The physical question of whether that object matches atomic reality remains open and is explicitly left to a deterministic executable to test.

/-- 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]

/-- 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)

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

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

/-- Known-wrong shortcut control: doubling only the unit-posting source
changes a diagonal matrix entry. The new carrier is therefore not insensitive
to its built-in dimensionless source normalization. -/
theorem doubled_source_decoy_changes_operator
    (op : CoherenceScaledEventOperator3) :
    ∃ i : Fin (TorusCard3 3),
      op.evolution.hamiltonian i i ≠
        (oneBodyEvolution3 eventGeneratorEnergyScale 1
          (q1AtomicModelPotential3 op.event
            (2 * eventGeneratorEnergyScale) 0)).hamiltonian i i := by
  obtain ⟨i, hi⟩ :=
    distinct_sourceDensityCoupling_distinct_diagonal
      op.event op.event_realized eventGeneratorEnergyScale 0
      eventGeneratorEnergyScale (2 * eventGeneratorEnergyScale)
      (by
        intro h
        linarith [eventGeneratorEnergyScale_pos])
  refine ⟨i, ?_⟩
  have hleft :
      op.evolution.hamiltonian i i =
        (oneBodyEvolution3 eventGeneratorEnergyScale 1
          (q1AtomicModelPotential3 op.event
            eventGeneratorEnergyScale 0)).hamiltonian i i := by
    rw [evolution_eq_old_interface_model,
      oneBodyEvolution3_depends_on_sourceDensityCoupling,
      oneBodyEvolution3_depends_on_sourceDensityCoupling]
    simp [sourceDensityCoupling]
    rw [eventGeneratorEnergyScale_eq_E_coh]
  intro hsame
  exact hi (hleft.symm.trans hsame)

What this page does not claim

The theorem does not prove that CoherenceScaledEventOperator3 is the molecular Hamiltonian. The theorem does not prove the physical correctness of the emitted spectrum. The theorem does not establish source-action duality or RemainingPhysicalEquality.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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