Foundation Pair Kernel Coherence Scaled Event Operator Coherence Born Potential

The linear response

The declaration coherenceBornPotentialEnergy3_line is a formal theorem inside the Recognition Science framework's machine-checked library of formal theorems. In plain language, it establishes that a particular energy expression, the Born potential energy for a three-dimensional model, changes linearly when you add a scaled variation to its input potential. If you write the potential as a base value plus a small change multiplied by a number t, the resulting energy equals the base energy plus t times a separate response term. This is the standard linearity property that any reasonable energy functional must satisfy.

The theorem is not about physics directly. It is a statement about a mathematical definition. The definition, bornPotentialEnergy3, takes a potential and a wavefunction, and returns the sum over all positions of the potential times the squared magnitude of the wavefunction. This is the familiar quantum-mechanical expectation value of the potential energy. The theorem proves that this expression is linear in the potential, with the response term being the expectation value of the variation itself. The proof is a direct computation, marked as a theorem in the library with no unproven assumptions.

In Recognition Science, this result is part of a larger construction. The framework builds a specific operator, the CoherenceScaledEventOperator3, meant to represent a single quantum particle. This operator carries no free parameters; its evolution is fixed by a single energy scale and a potential derived from a signed Green's function. The linearity theorem is one of several properties that the framework proves about this operator, alongside its Hermiticity and its Stone evolution property. These are mathematical guarantees that the operator behaves like a proper quantum Hamiltonian.

What the theorem does not claim is equally important. It does not prove that this operator is the actual Hamiltonian of a hydrogen atom or any other physical system. The framework explicitly labels this construction as a model, not a derivation. The choice of the shared unit-posting operator is new physical structure, and the definition fixes the dimensionless coupling ratio to one. The theorem says nothing about the physical correctness of the emitted spectrum, and it does not prove any equality with remaining physical quantities. The framework's own documentation states that a deterministic executable must now put this exact object in front of atomic reality to test it.

/-- Exact line response: varying the potential reads the Born density from
the same scalar energy. -/
theorem bornPotentialEnergy3_line
    (potential variation : PotentialVariation3)
    (ψ : OneBodyHilbert3) (t : ℝ) :
    bornPotentialEnergy3
        (fun i => potential i + t * variation i) ψ =
      bornPotentialEnergy3 potential ψ +
        t * bornSourceResponse3 ψ variation := by
  change
    (∑ i, (potential i + t * variation i) * Complex.normSq (ψ i)) =
      (∑ i, potential i * Complex.normSq (ψ i)) +
        t * ∑ i, variation i * Complex.normSq (ψ i)
  simp_rw [add_mul]
  rw [Finset.sum_add_distrib]
  apply congrArg₂ (· + ·) rfl
  calc
    (∑ i, t * variation i * Complex.normSq (ψ i)) =
        ∑ i, t * (variation i * Complex.normSq (ψ i)) := by
          apply Finset.sum_congr rfl
          intro i _
          ring
    _ = t * ∑ i, variation i * Complex.normSq (ψ i) := by
          rw [Finset.mul_sum]

/-- Exact line response: varying the potential reads the Born density from
the same scalar energy. -/
theorem bornPotentialEnergy3_line
    (potential variation : PotentialVariation3)
    (ψ : OneBodyHilbert3) (t : ℝ) :
    bornPotentialEnergy3
        (fun i => potential i + t * variation i) ψ =
      bornPotentialEnergy3 potential ψ +
        t * bornSourceResponse3 ψ variation := by
  change
    (∑ i, (potential i + t * variation i) * Complex.normSq (ψ i)) =
      (∑ i, potential i * Complex.normSq (ψ i)) +
        t * ∑ i, variation i * Complex.normSq (ψ i)
  simp_rw [add_mul]
  rw [Finset.sum_add_distrib]
  apply congrArg₂ (· + ·) rfl
  calc
    (∑ i, t * variation i * Complex.normSq (ψ i)) =
        ∑ i, t * (variation i * Complex.normSq (ψ i)) := by
          apply Finset.sum_congr rfl
          intro i _
          ring
    _ = t * ∑ i, variation i * Complex.normSq (ψ i) := by
          rw [Finset.mul_sum]

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

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

What this page does not claim

This theorem does not claim that the CoherenceScaledEventOperator3 is the Hamiltonian of hydrogen or any other atom. This theorem does not claim that the model's emitted spectrum is physically correct. This theorem does not claim to prove any equality with remaining physical quantities.

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

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