Foundation Pair Kernel Green3 Source Scale Changes Potential Coeff

Source scale and the potential coefficient

In classical physics, the potential around a point source is the product of two things: a fixed shape determined by the geometry of space, and a number that says how strong the source is. In three dimensions the shape is the familiar inverse distance, and the strength is the charge or mass. The theorem source_scale_changes_potentialCoeff isolates that second factor. It shows that two different source strengths, 1 and 2, give two different potential coefficients. That is all it shows, and the demonstration is a short algebraic check.

The statement lives in the Recognition Science framework, which models physical structure from a discrete record of events it calls a ledger. The framework's machine-checked library of formal theorems contains this result in the module PairKernelGreen3. The formal definition sets the potential coefficient to the source strength divided by 4π, the same geometric factor that appears in Coulomb's law and Newton's law. The theorem then verifies that the map from source strength to coefficient is injective: no two distinct strengths collapse to the same coefficient.

What the theorem does not do is more important than what it does. It does not show that the coefficient for a lattice of recognition events is actually 1/(4π). That target, named GreenCoefficientTarget3, is stated but not proved in the module. The file is explicit that the hard theorem remains open. It also does not identify the source strength with any physical quantity. A structure called GreenReadout3 carries a field named couplingScale, but the docstring says plainly that this value is uninterpreted and is not Newton's constant.

The module includes a deliberate decoy to show why the target alone cannot determine a coupling. A theorem shows that the same Green coefficient can accompany different source strengths and different coupling scales. In plain terms: knowing the asymptotic shape of a field does not tell you what produced it. The framework's own library demonstrates this limitation, and the theorem about source scale is the small piece that makes the point precise.

theorem source_scale_changes_potentialCoeff :
    sourceScaledPotentialCoeff 1 ≠ sourceScaledPotentialCoeff 2 := by
  exact sourceScaledPotentialCoeff_ne_of_ne (by norm_num)

/-- A free source strength multiplies the unit Green coefficient. -/
def sourceScaledPotentialCoeff (sourceScale : ℝ) : ℝ :=
  sourceScale / (4 * Real.pi)

/-- Green normalization alone leaves the source and coupling scales
unidentified. This is the R6 decoy gate. -/
theorem greenCoeff_alone_does_not_identify_source_or_coupling :
    ∃ R₁ R₂ : GreenReadout3,
      R₁.greenCoefficient = greenCoeff3 ∧
      R₂.greenCoefficient = greenCoeff3 ∧
      R₁.greenCoefficient = R₂.greenCoefficient ∧
      R₁.sourceScale ≠ R₂.sourceScale ∧
      R₁.couplingScale ≠ R₂.couplingScale := by
  refine
    ⟨{ greenCoefficient := greenCoeff3, sourceScale := 1, couplingScale := 1 },
      { greenCoefficient := greenCoeff3, sourceScale := 2, couplingScale := 2 },
      ?_⟩
  norm_num

What this page does not claim

The theorem does not claim that the 1/(4π) coefficient is proved for any lattice. It does not claim that the source strength is identified with charge or mass. It does not claim that couplingScale is Newton's constant.

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/PairKernelGreen3.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.

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