Encyclopedia Foundation Foundation Pair Kernel Coherence Scaled Event Operator Doubled Source Decoy Chan

ARTICLE 3 claims 3 theorems

Foundation Pair Kernel Coherence Scaled Event Operator Doubled Source Decoy Chan

In the framework's model of a single quantum event, doubling the strength of a decoy potential demonstrably changes the operator, so a decoy cannot be mistaken for the real source.

Verify this page

A source decoy is not the source

A ledger, a discrete record of events, is the starting point of Recognition Science. From this ledger the framework builds a model of a single quantum particle on a small three-site torus. The model carries one realized primitive posting, a single recorded event, and from that posting it constructs a finite one-body operator that governs the particle's evolution. The operator is born with a signed graph Laplacian, a mean-zero Green response, and one common generator scale, the native energy per fundamental tick. There are no free real coefficient fields on the carrier; the source response is the real-linear Born-density covector obtained by varying the same scalar potential energy.

The declaration doubled_source_decoy_changes_operator is a theorem in the machine-checked library of formal theorems. It states that if you take the model's operator and replace the source potential with a decoy potential whose strength is doubled, the resulting operator is different. Specifically, there exists at least one site on the torus where the diagonal Hamiltonian entry, the energy at that site, differs between the original and the decoy operator. The theorem is proved in the library with no sorry and no new axiom.

What does this establish in plain terms? It establishes that a decoy potential, one that mimics the source but with a different strength, is not equivalent to the source. The model's operator is sensitive to the distinction. This is a coherence check: the framework's construction does not collapse into a trivial object where any potential looks the same. The decoy is distinguished from the source by the operator itself.

In Recognition Science, this theorem is a property of a MODEL carrier, not a proof of physical reality. The carrier is a definitional choice: the shared unit-posting operator is new physical structure. The current Recognition premises do not prove that this model is the molecular Hamiltonian, and the definition fixes its dimensionless coupling ratio to one. The theorem does not prove RemainingPhysicalEquality, source-action duality, or the physical correctness of the emitted spectrum. A deterministic executable must now put this exact object in front of atomic reality.

THEOREM doubled_source_decoy_changes_operator · IndisputableMonolith/Foundation/PairKernelCoherenceScaledEventOperator.lean
doubled_source_decoy_changes_operator · IndisputableMonolith/Foundation/PairKernelCoherenceScaledEventOperator.lean:285 
/-- 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)
THEOREM doubled_source_decoy_changes_operator · IndisputableMonolith/Foundation/PairKernelCoherenceScaledEventOperator.lean
doubled_source_decoy_changes_operator · IndisputableMonolith/Foundation/PairKernelCoherenceScaledEventOperator.lean:285 
/-- 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)
THEOREM doubled_source_decoy_changes_operator · IndisputableMonolith/Foundation/PairKernelCoherenceScaledEventOperator.lean
doubled_source_decoy_changes_operator · IndisputableMonolith/Foundation/PairKernelCoherenceScaledEventOperator.lean:285 
/-- 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 the model is the molecular Hamiltonian. The theorem does not prove source-action duality. The theorem 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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND