Foundation Dalembert Factorization Forcing Factorization Associativity Gate

The factorization gate

The FactorizationAssociativityGate is a precise mathematical condition on a two-input function P(u, v). It demands four properties: symmetry (swapping the inputs changes nothing), affine response in the second input (for fixed u, P(u, v) is a straight line in v), a boundary law P(u, 0) = 2u, and the normalization P(1, 1) = 6. These are not arbitrary choices; they are the algebraic residue left after a harder analytic step, which the framework's B2 closure program treats as already done.

What the gate establishes is a uniqueness result. If a function satisfies all four conditions, then it must be exactly the polynomial P(u, v) = 2uv + 2u + 2v. The framework proves this in two stages. First, symmetry plus the boundary law force the function to lie in a bilinear family: P(u, v) = c·uv + 2u + 2v for some constant c. Second, the normalization P(1, 1) = 6 forces c = 2. The final theorem states the equivalence: a function satisfies the gate if and only if it equals this canonical polynomial. The proof is machine-checked in the framework's library of formal theorems.

In Recognition Science, this polynomial is the RCL combiner, the rule that merges two recognition events into one. The gate does not derive the combiner from first principles; it takes the affine-response step as given and shows what algebra must follow. The framework's language is careful here: the gate is a packaged bridge, not a standalone discovery.

The practical consequence is a clean dichotomy. Either a combiner satisfies the four gate conditions and is the RCL polynomial, or it fails at least one condition and is something else. This is what the declaration establishes. It does not claim that every physically meaningful combiner must satisfy the gate, nor does it derive the affine-response step itself. The gate is a conditional result: if the analytic precondition holds, the algebraic conclusion is forced.

/-- Canonical normalization selects the RCL member of the bilinear family. -/
theorem gate_forces_rcl (P : ℝ → ℝ → ℝ)
    (hGate : FactorizationAssociativityGate P) :
    ∀ u v, P u v = 2 * u * v + 2 * u + 2 * v := by
  obtain ⟨c, hc⟩ := gate_forces_bilinear_family P hGate
  have hc_two : c = 2 := by
    have h11 : P 1 1 = c * 1 * 1 + 2 * 1 + 2 * 1 := by
      simpa using hc 1 1
    linarith [hGate.unitDiagonal, h11]
  intro u v
  calc
    P u v = c * u * v + 2 * u + 2 * v := hc u v
    _ = 2 * u * v + 2 * u + 2 * v := by rw [hc_two]

/-- Exact gate characterization: the factorization gate is equivalent to being
the canonical RCL combiner. -/
theorem factorization_gate_iff_rcl (P : ℝ → ℝ → ℝ) :
    FactorizationAssociativityGate P ↔ ∀ u v, P u v = rclCombiner u v := by
  constructor
  · intro hGate u v
    rw [gate_forces_rcl P hGate u v]
    rfl
  · intro hP
    refine {
      symmetric := ?_
      rightAffine := ?_
      zeroBoundary := ?_
      unitDiagonal := ?_
    }
    · intro u v
      rw [hP u v, hP v u]
      unfold rclCombiner
      ring
    · intro u
      refine ⟨2 * u + 2, 2 * u, ?_⟩
      intro v
      rw [hP u v]
      unfold rclCombiner
      ring
    · intro u
      rw [hP u 0]
      unfold rclCombiner
      ring
    · rw [hP 1 1]
      unfold rclCombiner
      norm_num

What this page does not claim

The gate does not derive the affine-response step; it assumes it. The gate does not claim every two-input combiner must satisfy its four conditions. The gate does not establish the physical interpretation of the RCL combiner.

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/DAlembert/FactorizationForcing.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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