Foundation Pair Kernel Bounded Coupling Lattice3 Recognition Relation Bounded

Bounded recognition in three dimensions

A recognition relation is a rule that says which pairs of objects in a system are allowed to interact. In the Recognition Science framework, the ledger, a discrete record of events, carries such a relation on every tick. A relation is bounded when every interacting pair lies within a fixed distance of each other, measured by some metric on the index set. The theorem lattice3RecognitionRelation_bounded proves, for a three-dimensional cubic lattice of side length L, that its recognition relation is bounded with radius 1 under the natural encoded ℓ¹ distance. In plain words: on this lattice, only nearest neighbors, objects one step apart, are recognized as related, and nothing farther away is.

The proof is short because the relation is defined to be exactly that: two positions are related precisely when their encoded distance is at most 1. The theorem is a direct unpacking of the definition, not a deep derivation. Its value is architectural. It supplies a concrete, machine-checked example of a bounded recognition relation in three dimensions, which the framework's library of formal theorems can use as a building block. This matters because the framework aims to show that three-dimensional spatial structure can arise from recognition dynamics, and this theorem provides a clean, verified instance of a three-dimensional lattice whose interactions are local.

In Recognition Science, this theorem is part of a larger story about what bare recognition dynamics can and cannot force. A companion theorem proves a negative result: bare recognition structure plus a valid atomic tick does not force bounded coupling. There exist carriers whose recognition relation is the all-true relation, meaning every pair interacts, and which therefore fail to be bounded. The three-dimensional theorem is the positive counterpart: under a committed lattice geometry, boundedness holds. The framework labels this geometry as a chosen model, not as something derived from deeper principles. Forcing that commitment from recognition dynamics alone remains an open hypothesis, named ProductionLatticeCouplingObligation.

The theorem also enables a concrete consequence for the framework's weight graphs. A weight graph assigns interaction strengths to pairs. The theorem boxWeight_supported_on_lattice3 shows that a particular weight assignment, boxWeight, has nonzero weights only on pairs recognized by the lattice relation. Combined with the boundedness theorem, this yields boxWeight_finiteRangeOn_export_v1: the weight graph has finite range, meaning interactions vanish beyond distance 1. This is a theorem-grade export for the three-dimensional path, in contrast to the one-dimensional case where finite range remains a hypothesis. The framework thus has a verified, local, three-dimensional lattice model, while honestly marking what it has not yet forced.

The theorem does not claim that three-dimensional space itself is forced by recognition dynamics. It does not claim that the lattice geometry is the only possible one, nor that production weights in the real world must follow it. The boundedness is a property of a chosen model, not a consequence of bare recognition structure. The framework's own documentation is explicit: renaming this inhabitant as forced physics would be adverse. The theorem is a precise, verified statement about a specific construction, and its power lies in being exactly that, no more and no less.

theorem lattice3RecognitionRelation_bounded (L : ℕ) :
    BoundedRecognitionRelationOn
      (lattice3RecognitionRelation L) (encodedDist3 L) 1 := by
  intro i j hR
  exact hR

/-- 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⟩

theorem boxWeight_supported_on_lattice3 (L : ℕ) :
    RelationSupportsWeightOn (lattice3RecognitionRelation L) (boxWeight L) := by
  intro i j hne
  dsimp [lattice3RecognitionRelation, encodedDist3]
  change (if dist3 ((enc3 L).symm i) ((enc3 L).symm j) ≤ 1 then (1 : ℝ) else 0) ≠ 0 at hne
  split_ifs at hne with hle
  · exact hle
  · exact (hne rfl).elim

/-- **D=3 inhabitant of the metric export.** Under the committed lattice
geometry, nonzero box weights force `FiniteRangeOn` at radius 1. -/
theorem boxWeight_finiteRangeOn_export_v1 (L : ℕ) :
    BoundedRecognitionRelationOn
        (lattice3RecognitionRelation L) (encodedDist3 L) 1 ∧
      RelationSupportsWeightOn (lattice3RecognitionRelation L) (boxWeight L) ∧
        FiniteRangeOn (boxWeight L) (encodedDist3 L) 1 :=
  ⟨lattice3RecognitionRelation_bounded L,
    boxWeight_supported_on_lattice3 L,
    finiteRangeOn_export_v1
      (lattice3RecognitionRelation_bounded L)
      (boxWeight_supported_on_lattice3 L)⟩

What this page does not claim

The theorem does not force three-dimensional space from bare recognition dynamics. The theorem does not claim the lattice geometry is unique or physically necessary. The theorem does not establish that production weights in the real world follow this bounded relation.

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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