Foundation Pair Kernel Zero Fit One Body Atomic Checkpoint Fd2 One Body Coupling

The coupling ratio

In classical physics, the choice of units is a freedom, not a law. You can measure energy in joules, electronvolts, or ergs, and the physics does not change. The declaration oneBodyCouplingRatio_common_rescale, a theorem in the machine-checked library of formal theorems, states the exact consequence of that freedom for a finite one-body quantum model: when you multiply both the kinetic scale and the potential scale by the same nonzero number, the ratio of potential scale to kinetic scale stays exactly the same. In symbols, if the kinetic scale is K and the potential scale is P, then the coupling ratio P/K is unchanged when both are multiplied by a common factor c. The proof is a direct algebraic calculation, and the theorem carries the condition that neither scale is zero, because the ratio would otherwise be undefined.

The declaration's importance is that it identifies a dimensionless quantity that is invariant under a common energy-unit rescaling. This is not a claim about any specific physical value. The theorem does not say what the coupling ratio is, only that it is well-defined under the rescaling. It also does not say that the potential scale is the electron potential, that the kinetic scale is the electron kinetic energy, or that the ratio equals the fine-structure constant or any other measured constant. The theorem is a statement about the algebraic structure of the model, not about its physical interpretation.

In Recognition Science, the declaration sits inside a larger checkpoint module that builds a finite one-body quantum interface. The module attaches a specific attractive potential, the q=1 Green field, as a model choice, and it proves that the resulting Hamiltonian is Hermitian and generates a unitary evolution. But the module explicitly does not prove that this field is an electron potential, does not select the coupling ratio, and does not identify the framework's alpha expression with the physical infrared coupling. A separate predicate, RecognitionOneBodyQuantumAttachment3, states that missing semantic arrow as an unproved assumption. The coupling ratio theorem is therefore a clean algebraic fact, and the physical meaning of the ratio remains an open target.

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 potential scale is the electron potential. The theorem does not claim that the coupling ratio equals the fine-structure constant. The theorem does not claim that any physical Hamiltonian is identified with the model.

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)

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: