Foundation Pair Kernel Coherence Scaled Event Operator Born Potential Energy3 Eq

What the equality establishes

The declaration bornPotentialEnergy3_eq_sourceResponse proves a basic identity about how a quantum system responds to a small change in its environment. In plain terms, it says that the change in a system's energy when you slightly alter the potential it sits in is exactly the same as the response of its wavefunction to that alteration. This is a formal, machine-checked statement within the Recognition Science framework, and it holds for a specific model of a single particle on a three-site lattice.

The identity is a mathematical tautology in the framework's construction. It states that the Born potential energy (the energy of a state in a given potential, computed as the sum over sites of the potential times the probability density) equals the source response (the linear response of the state to a potential variation). The proof is by definitional equality: both expressions reduce to the same sum. This is not a physical discovery but a structural consistency check, ensuring that two different ways of computing the same quantity agree.

The identity is one of several properties proved for the CoherenceScaledEventOperator3, a model of a single quantum event. The framework also proves that this operator is Hermitian, that its evolution satisfies Stone's theorem, and that its source response is linear. These are all formal properties of the model, not empirical claims about nature.

What the declaration does not claim is more important than what it does. It does not prove that this model is the actual Hamiltonian of a physical molecule. It does not establish that the emitted spectrum matches any measured atomic spectrum. It does not derive the fine-structure constant or any other physical constant from first principles. The model fixes its dimensionless coupling ratio to one, a choice, not a derivation. The framework's own documentation states this plainly: the model is a carrier with theorem properties, not a proven physical law.

In the broader context of Recognition Science, this identity is a small but necessary link in a chain. It ensures that the framework's internal language is coherent: energy and response are not separate concepts but two views of the same underlying structure. The next step, as the framework's authors note, is to test this exact object against atomic reality, a task that remains open.

theorem bornPotentialEnergy3_eq_sourceResponse
    (potential : PotentialVariation3)
    (ψ : OneBodyHilbert3) :
    bornPotentialEnergy3 potential ψ =
      bornSourceResponse3 ψ potential :=
  rfl

theorem bornPotentialEnergy3_eq_sourceResponse
    (potential : PotentialVariation3)
    (ψ : OneBodyHilbert3) :
    bornPotentialEnergy3 potential ψ =
      bornSourceResponse3 ψ potential :=
  rfl

/-- 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)

What this page does not claim

The model is not proven to be the molecular Hamiltonian. The identity does not derive any physical constant such as the fine-structure constant. The emitted spectrum is not proven to match any measured atomic 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

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