Foundation Pair Kernel Canonical Generator Source S9 Recognition Green Consumer

Two Green scales

A Green's function is a standard tool in physics and mathematics: it is the response of a system to a single, sharp input, like the ripple from a stone dropped in still water. On a discrete space, such as a lattice or a torus, the input is a posting at one site and a removal at another, a dipole. The framework's declaration recognitionGreenConsumer_admits_scale_one_and_two proves that on a torus with at least two sites, two different Green's functions, one scaled by 1 and one by 2, both satisfy the same defining equation: the Laplacian of the response equals the source dipole. It also proves these two responses are not the same function.

The result is a statement about the framework's discrete geometry, not about any particular physical system. The framework models recognition events as postings in a ledger, a discrete record of events, and the theorem shows that the cost of a posting, its magnitude, can be 1 or 2 while still generating a valid response. This is a structural fact: the equation that defines a Green's function does not pick a unique scale for the source. The theorem is proved in the machine-checked library of formal theorems, with no unproved assumptions beyond the standard logical axioms.

The theorem does not claim that these are the only possible scales, nor does it assign a physical meaning to the numbers 1 and 2. It does not say that a posting of magnitude 2 is twice as energetic as one of magnitude 1, only that both satisfy the same formal equation. The result is a building block for later work that attaches physical units to ledger postings; that conversion remains an explicit parameter, not a consequence of this theorem.

/-- Green existence and neutrality alone do not select scale: the same
constructed generator compiles at scales one and two with distinct sources. -/
theorem recognitionGreenConsumer_admits_scale_one_and_two
    {N : ℕ} [NeZero N]
    (hN : 2 ≤ N)
    (σ : Equiv.Perm (Fin 3))
    (a b p : TorusSite3 N)
    (hab : a ≠ b) :
    framedTorusLaplacian σ
        (scaledDipoleGreenResponse 1 a b) p =
          scaledTorusDipole 1 a b p ∧
      framedTorusLaplacian σ
        (scaledDipoleGreenResponse 2 a b) p =
          scaledTorusDipole 2 a b p ∧
      scaledTorusDipole 1 a b ≠ scaledTorusDipole 2 a b := by
  refine
    ⟨(recognitionConstruction_globalGreen_consumer
        hN σ 1 a b p).2.2,
      (recognitionConstruction_globalGreen_consumer
        hN σ 2 a b p).2.2,
      ?_⟩
  intro h
  have ha := congrFun h a
  simp [scaledTorusDipole, hab] at ha

/-- Green existence and neutrality alone do not select scale: the same
constructed generator compiles at scales one and two with distinct sources. -/
theorem recognitionGreenConsumer_admits_scale_one_and_two
    {N : ℕ} [NeZero N]
    (hN : 2 ≤ N)
    (σ : Equiv.Perm (Fin 3))
    (a b p : TorusSite3 N)
    (hab : a ≠ b) :
    framedTorusLaplacian σ
        (scaledDipoleGreenResponse 1 a b) p =
          scaledTorusDipole 1 a b p ∧
      framedTorusLaplacian σ
        (scaledDipoleGreenResponse 2 a b) p =
          scaledTorusDipole 2 a b p ∧
      scaledTorusDipole 1 a b ≠ scaledTorusDipole 2 a b := by
  refine
    ⟨(recognitionConstruction_globalGreen_consumer
        hN σ 1 a b p).2.2,
      (recognitionConstruction_globalGreen_consumer
        hN σ 2 a b p).2.2,
      ?_⟩
  intro h
  have ha := congrFun h a
  simp [scaledTorusDipole, hab] at ha

What this page does not claim

The theorem does not claim these are the only possible scales for a Green's function. It does not assign a physical meaning to the numbers 1 and 2. It does not say that a posting of magnitude 2 is twice as energetic as one of magnitude 1.

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/PairKernelCanonicalGeneratorSourceS9.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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