Foundation Pair Kernel Bounded Coupling Mean Field Weight Not Finite Range On En

The mean-field limit

In the Recognition Science framework, a ledger (a discrete record of recognition events) can carry weights on the links between its entries. The mean-field weight is the simplest possible rule: every entry links to every other entry with the same strength. The declaration meanFieldWeight_not_finiteRangeOn_encodedDist3 is a proved theorem stating that this uniform rule cannot be a local rule in three dimensions. Concretely, on a three-dimensional lattice of size L by L by L (with L at least 2), the mean-field weight assigns a nonzero value to pairs of points that are far apart, so it fails the condition of finite range.

Finite range is the formal way of saying that interactions are local: only entries within a small distance of each other can influence each other. The proof works by measuring distance with the standard "city-block" metric on the lattice, where moving one step along any axis counts as distance 1. The theorem shows that the mean-field weight connects points at distance greater than 1, so it cannot be a nearest-neighbor interaction. This is not a surprise: a rule that connects everything to everything is the opposite of a local rule. The value of the theorem is that it is machine-checked, meaning the proof is verified by a computer program that checks every logical step.

The theorem also contrasts with a positive result in the same file. A different weight, called box weight, assigns nonzero values only to nearest neighbors on the lattice. The framework proves that this box weight does satisfy the finite-range condition. So the framework does not claim that locality is impossible; it claims that the specific mean-field rule is not local, while a specific local alternative exists.

What the theorem does not claim is broader. It does not prove that the mean-field weight is the only non-local rule, nor that locality is forced by the framework's basic assumptions. The framework's basic structure plus a valid tick (a single recognition event) can coexist with weights that are not local at all. The theorem only states a fact about one particular weight on one particular geometry. The framework also does not claim that the box weight is the physically correct rule; that remains an open question, named in the file as ProductionLatticeCouplingObligation.

The practical consequence is a precise boundary. If the framework is to describe a three-dimensional world with local interactions, the mean-field rule cannot be the production rule. The framework must either find a deeper reason that forces local weights, or commit to a specific local rule like the box weight. The theorem sharpens the question: it says where the open problem lies, without pretending to solve it.

theorem meanFieldWeight_not_finiteRangeOn_encodedDist3
    (L : ℕ) (hL : 2 ≤ L) :
    ¬ FiniteRangeOn (meanFieldWeight (L * L * L)) (encodedDist3 L) 1 := by
  intro hFR
  -- Two axis steps (still available at L = 2) give encodedDist3 = 2 > 1.
  set o : Site3 L := ((⟨0, by omega⟩, ⟨0, by omega⟩), ⟨0, by omega⟩)
  set e : Site3 L := ((⟨1, by omega⟩, ⟨1, by omega⟩), ⟨0, by omega⟩)
  have hdist : encodedDist3 L (enc3 L o) (enc3 L e) = 2 := by
    simp [encodedDist3, Equiv.symm_apply_apply, dist3, Nat.dist, o, e]
  have hfar : 1 < encodedDist3 L (enc3 L o) (enc3 L e) := by
    rw [hdist]
    norm_num
  have hz := hFR (enc3 L o) (enc3 L e) hfar
  simp only [meanFieldWeight] at hz
  exact one_ne_zero hz

/-- Direct form: `boxWeight` is finite-range in the encoded ℓ¹ metric. -/
theorem boxWeight_finiteRangeOn (L : ℕ) :
    FiniteRangeOn (boxWeight L) (encodedDist3 L) 1 :=
  (boxWeight_finiteRangeOn_export_v1 L).2.2

/-- Bundle alias matching the named season obligation string. -/
theorem BoundedRecognitionCouplingObligation_not_forced_by_bare_RS
    (radius : ℕ) :
    ∃ n : ℕ,
      Nonempty (AtomicTick (tickCarrier n)) ∧
        ¬ BoundedRecognitionRelation (tickCarrier n).R radius ∧
          ¬ FiniteRange (meanFieldWeight n) radius := by
  rcases recognitionStructure_atomicTick_do_not_force_boundedCoupling radius with
    ⟨n, hTick, hB, _hS, hFR⟩
  exact ⟨n, hTick, hB, hFR⟩

What this page does not claim

The mean-field weight is the only non-local weight; the theorem concerns one specific weight. Locality is forced by the framework's basic assumptions; the theorem shows the opposite. The box weight is the physically correct rule; it is a committed geometry, not a derived one.

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/PairKernelBoundedCoupling.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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