Foundation Pair Kernel Green3 Green Normalization3

The normalization package

A Green function is a mathematical object that tells how a field spreads from a point source; in three dimensions, the familiar 1/r potential is its asymptotic shape. The Recognition Science declaration GreenNormalization3 packages the data needed to study such a function on a discrete cubic lattice: the Green readout itself, a source strength, and a potential coefficient. The package's defining equation ties the potential coefficient to the source strength by dividing by 4π, the standard three-dimensional geometric factor. The declaration establishes that distinct source strengths produce distinct potential coefficients, a fact that sounds trivial but carries a consequence.

That consequence is the package's real content. The declaration establishes a decoy theorem: knowing only the asymptotic coefficient 1/(4π) does not identify the source strength or any coupling scale. Two different readouts can share the same coefficient while their source scales differ. In plain terms, matching the expected 1/r tail of a three-dimensional field is not enough to pin down how strong the source is. The framework's machine-checked library of formal theorems records this as a deliberate gate, separating what the target states from what the source normalization can determine.

The package also names the exact target that remains unproved. It defines a predicate asserting that a given Green function has the asymptotic coefficient 1/(4π), and attaches that predicate to the package as an obligation. No declaration in the file establishes that the cubic recognition lattice actually achieves this coefficient. The package is inhabited by a dummy readout that makes no claim to be a lattice Green function, so the structure exists without asserting the hard theorem. The file's own documentation says the hard theorem is still open.

What the declaration does not claim is as important as what it establishes. It does not establish that the three-dimensional coefficient 1/(4π) emerges from the recognition lattice. It does not identify the source strength or any coupling scale, and the decoy theorem shows why: the asymptotic coefficient alone leaves a free scale. It contains no gravity, Planck, SI, or Newton-coupling content, and the coupling scale in the decoy structure is explicitly uninterpreted, not Newton's constant. The package is a precise boundary marker: it separates the target from the normalization, establishes that the separation matters, and leaves the target as a named open problem.

/-- Distinct source strengths give distinct potential coefficients. -/
theorem sourceScaledPotentialCoeff_ne_of_ne {S₁ S₂ : ℝ} (hS : S₁ ≠ S₂) :
    sourceScaledPotentialCoeff S₁ ≠ sourceScaledPotentialCoeff S₂ := by
  unfold sourceScaledPotentialCoeff
  intro h
  have hden : (4 : ℝ) * Real.pi ≠ 0 := by positivity
  have hmul := congrArg (fun x : ℝ => x * (4 * Real.pi)) h
  field_simp [hden] at hmul
  exact hS hmul

/-- 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

/-- The package type is inhabited without asserting the hard target. The
dummy readout carries no claim of being a lattice Green function. -/
def unitSourceInterface : GreenNormalization3 where
  green := fun _ => 0
  sourceScale := 1
  sourceScale_ne_zero := by norm_num
  potentialCoefficient := sourceScaledPotentialCoeff 1
  potentialCoefficient_eq_source_scale := rfl

/-- A normalization package keeps the Green readout and source strength
separate. The future hard theorem is `GreenNormalization3.target`. -/
structure GreenNormalization3 where
  green : ℕ → ℝ
  sourceScale : ℝ
  sourceScale_ne_zero : sourceScale ≠ 0
  potentialCoefficient : ℝ
  potentialCoefficient_eq_source_scale :
    potentialCoefficient = sourceScaledPotentialCoeff sourceScale

What this page does not claim

GreenNormalization3 does not establish the 1/(4π) coefficient for the recognition lattice. The declaration does not identify the source strength or any coupling scale. The couplingScale in the decoy structure is not 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.

Derived articles

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