Foundation Pair Kernel Green3 Green Asymptotic Coefficient3

The asymptotic target

A Green function is a mathematical tool that describes how an influence spreads from a source, like ripples from a stone dropped in water. In three dimensions, the influence from a point source typically falls off with distance, and the product of distance and influence settles toward a constant value. The declaration GreenAsymptoticCoefficient3 defines precisely what it means for such a function to have a specific asymptotic coefficient: for any small tolerance, beyond some radius the product of radius and the function's value stays within that tolerance of the target coefficient. The target coefficient itself is defined as 1 divided by 4 pi, the same constant that appears in the classical Coulomb and gravitational potentials.

The declaration is a target, not a proof. The machine-checked library of formal theorems states this target for a three-dimensional recognition lattice, a discrete grid of points where the framework's recognition events are recorded, but it does not prove that the lattice's Green function actually reaches the coefficient 1 over 4 pi. That remains an open problem. What the library does prove is a decoy: the same Green coefficient can accompany completely different source strengths and coupling scales. The theorem greenCoeff_alone_does_not_identify_source_or_coupling constructs two readouts with identical Green coefficients but different source and coupling scales, showing that the asymptotic coefficient by itself cannot determine a Newton-type coupling.

This separation matters because it keeps the target honest. The framework models a normalization package that keeps the Green readout and the source strength separate, with the potential coefficient defined as the source scale divided by 4 pi. A proved theorem shows that distinct source strengths always give distinct potential coefficients, so the source scale is not a free ambiguity once the potential coefficient is fixed. But the source scale itself remains uninterpreted: the library explicitly states that the coupling scale in its minimal readout is not Newton's constant. The declaration establishes what the target is, and the decoy establishes what the target alone cannot do.

/-- A radial Green readout has asymptotic coefficient `C` when `r * G(r)`
converges to `C` along the natural-number shells. -/
def GreenAsymptoticCoefficient3 (green : ℕ → ℝ) (C : ℝ) : Prop :=
  ∀ ε : ℝ, 0 < ε → ∃ R : ℕ, ∀ r : ℕ, R ≤ r →
    |(r : ℝ) * green r - C| < ε

/-- The D=3 continuum Green coefficient targeted by R6. -/
def greenCoeff3 : ℝ := 1 / (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

/-- 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

What this page does not claim

The declaration does not prove that the lattice Green function reaches the coefficient 1 over 4 pi. The declaration does not identify the source scale or coupling scale as Newton's constant. The declaration does not derive the fine-structure constant or any other physical coupling 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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