Foundation Dalembert Factorization Forcing Gate Forces Rcl

The factorization gate

A combining operation takes two numbers and returns a third, like addition or multiplication. In the Recognition Science framework, such an operation is called a combiner, a rule that merges two inputs into one output. The framework asks a narrow question: if a combiner obeys four plain conditions, what must its formula be? The answer, proved in the machine-checked library of formal theorems, is that the formula is forced to be exactly P(u,v) = 2uv + 2u + 2v.

The four conditions are the gate. The combiner must be symmetric, so P(u,v) = P(v,u). It must be affine in its second argument, meaning for each fixed u the output is a straight line in v. It must satisfy the boundary law P(u,0) = 2u. And it must meet the normalization P(1,1) = 6. These are not arbitrary; they encode the framework's ideas of symmetry, a zero-cost boundary, and a canonical unit scale. The theorem gate_forces_rcl shows that any real-valued combiner passing all four conditions is identically the formula above, with no free parameters left over.

The proof runs in two stages. First, symmetry plus the boundary law force the combiner into a bilinear family: for some constant c, P(u,v) = cuv + 2u + 2v. The normalization at (1,1) then pins c to 2, leaving the unique RCL combiner. A companion theorem states the equivalence exactly: a combiner satisfies the gate if and only if it equals this canonical formula. The framework's library records these as formal theorems, checked by a machine, with no hidden assumptions beyond the standard logical axioms.

In Recognition Science, this gate is the algebraic core of a larger closure program. It shows that the framework's combining rule is not chosen freely; once the four conditions are accepted, the polynomial follows. The result is a stepping stone, not a destination. It justifies the specific form of the RCL combiner used elsewhere in the framework, but it says nothing about why those four conditions should hold physically. That bridge, from recognition events to this algebraic structure, remains open.

What the gate does not claim is just as important. It does not derive the four conditions from anything deeper; they are assumed as the gate's input. It does not establish that the combiner is affine in its first argument, only in its second. And it does not connect the polynomial to any measured physical quantity. The theorem is a pure algebraic implication: if the gate, then the formula. The framework's larger claims about physics rest on additional steps beyond this one.

/-- 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]

/-- Once the affine-response step is known, symmetry and the boundary law force
    the entire bilinear family. -/
theorem gate_forces_bilinear_family (P : ℝ → ℝ → ℝ)
    (hGate : FactorizationAssociativityGate P) :
    ∃ c : ℝ, ∀ u v, P u v = c * u * v + 2 * u + 2 * v := by
  classical
  choose α β hAffine using hGate.rightAffine
  have hβ : ∀ u, β u = 2 * u := by
    intro u
    have h0 : P u 0 = α u * 0 + β u := hAffine u 0
    rw [hGate.zeroBoundary u] at h0
    linarith
  let c : ℝ := α 1 - 2
  refine ⟨c, ?_⟩
  intro u v
  have hsym1 : P u 1 = P 1 u := hGate.symmetric u 1
  have hαu : α u = c * u + 2 := by
    dsimp [c]
    have hcalc : α u * 1 + β u = α 1 * u + β 1 := by
      calc
        α u * 1 + β u = P u 1 := by symm; exact hAffine u 1
        _ = P 1 u := hGate.symmetric u 1
        _ = α 1 * u + β 1 := hAffine 1 u
    rw [hβ u, hβ 1] at hcalc
    linarith
  calc
    P u v = α u * v + β u := hAffine u v
    _ = (c * u + 2) * v + 2 * u := by rw [hαu, hβ u]
    _ = c * u * v + 2 * u + 2 * v := by ring

/-- 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 four gate conditions are not derived from anything deeper; they are assumed as input. The theorem does not show the combiner is affine in its first argument, only in its second. No connection is made between the RCL polynomial and any measured physical quantity.

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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