Foundation Pair Kernel Zero Fit One Body Atomic Checkpoint Fd2 One Body Scalar E

Energy rescaling

In the Recognition Science framework, a finite one-body quantum model on a discrete torus carries a scalar energy function. The theorem oneBodyScalarEnergy3_common_rescale states that if you multiply both the kinetic and potential scale parameters by a common factor c, the entire scalar energy is multiplied by c. In symbols: E(c·K, c·V, φ) = c·E(K, V, φ). This is the statement that the energy is homogeneous of degree one under a common rescaling of its two scale parameters.

This is a purely algebraic identity. It follows directly from the definition of the energy as a sum of a kinetic term and a potential term, each linear in its own scale. The theorem does not require any physical interpretation of the scales, any particular form of the potential, or any choice of units. It holds for any real numbers c, K, V and any real-valued state and potential functions on the finite torus.

The same module also proves that the dimensionless ratio V/K is invariant under this common rescaling. This ratio, called oneBodyCouplingRatio, survives a change of energy units. The theorem states that if c is nonzero and K is nonzero, then (c·V)/(c·K) = V/K. This is a simple consequence of the algebraic identity, but it is the reason the ratio is a meaningful quantity to study.

In Recognition Science, this theorem is part of a larger check that the finite one-body quantum interface is consistent. The module explicitly does not claim that the q=1 Green potential is an electron potential, does not select the kinetic-to-field coupling ratio, does not identify the current alpha expression with the physical infrared coupling, and does not select SI units. The theorem establishes only the algebraic rescaling property, not any physical content.

/-- A common energy-unit rescaling multiplies the entire scalar energy. -/
theorem oneBodyScalarEnergy3_common_rescale
    (c kineticScale potentialScale : ℝ)
    (potential state : Fin (TorusCard3 3) → ℝ) :
    oneBodyScalarEnergy3 (c * kineticScale) (c * potentialScale)
        potential state =
      c * oneBodyScalarEnergy3 kineticScale potentialScale potential state := by
  unfold oneBodyScalarEnergy3
  ring

theorem oneBodyCouplingRatio_common_rescale
    (c kineticScale potentialScale : ℝ)
    (hc : c ≠ 0) (hk : kineticScale ≠ 0) :
    oneBodyCouplingRatio (c * kineticScale) (c * potentialScale) =
      oneBodyCouplingRatio kineticScale potentialScale := by
  unfold oneBodyCouplingRatio
  field_simp

What this page does not claim

The theorem does not claim that the q=1 Green potential is an electron potential. The theorem does not select the kinetic-to-field coupling ratio. The theorem does not identify the current alpha expression with the physical infrared coupling.

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/PairKernelZeroFitOneBodyAtomicCheckpointFD2.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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