Foundation Dalembert Entanglement Gate Separable Implies Zero Mixed Diff

The separability test

A function of two variables is separable when its value at a pair of inputs is the sum of two single-variable functions, one for each input. The classical example is P(u, v) = 2u + 2v, which splits into α(u) = 2u and β(v) = 2v. Such functions have a defining property: the mixed second difference, P(u₁, v₁) - P(u₁, v₀) - P(u₀, v₁) + P(u₀, v₀), is always zero. This is the discrete analogue of the mixed partial derivative ∂²P/∂u∂v vanishing, and it holds for every separable function, not just the linear example.

The theorem separable_implies_zero_mixed_diff in the framework's machine-checked library of formal theorems proves exactly this: if P is separable, then that mixed difference is identically zero. The proof is a short algebraic manipulation using the definition of separability. The converse also matters: a function whose mixed difference is never zero cannot be separable. The framework's combiner P(u, v) = 2uv + 2u + 2v has mixed difference 2(u₁ - u₀)(v₁ - v₀), which is nonzero whenever the inputs differ, so it is not separable. The 2uv term is the interaction that couples the two observations.

In Recognition Science, the framework models the cost of recognizing a composite system xy as a function P of the costs of recognizing x and y separately. A separable P means the composite cost is just the sum of the parts, with no interaction. The framework calls this state non-entangling and proves that separability implies it. The theorem separable_implies_not_entangling states this directly. The framework's entanglement gate is the requirement that P have a nonzero cross-derivative, and it proves that the additive combiner fails this test while the RCL combiner passes it.

The declaration does not claim that every function with zero mixed difference is separable, nor does it say anything about quantum mechanical entanglement. It is a purely algebraic statement about real-valued functions of two real variables. The framework's use of the word entanglement is a deliberate analogy to quantum systems, but the theorem itself concerns only the arithmetic of the combiner, not the state space of a physical system.

/-- Separable implies zero mixed difference. -/
theorem separable_implies_zero_mixed_diff (P : ℝ → ℝ → ℝ) (hSep : IsSeparable P) :
    ∀ u₀ v₀ u₁ v₁, P u₁ v₁ - P u₁ v₀ - P u₀ v₁ + P u₀ v₀ = 0 := by
  obtain ⟨α, β, h⟩ := hSep
  intro u₀ v₀ u₁ v₁
  simp only [h]
  ring

/-- Separable implies not entangling (contrapositive of entangling implies not separable). -/
theorem separable_implies_not_entangling (P : ℝ → ℝ → ℝ) (hSep : IsSeparable P) :
    ¬ IsEntangling P := by
  intro ⟨u₀, v₀, u₁, v₁, h⟩
  exact h (separable_implies_zero_mixed_diff P hSep u₀ v₀ u₁ v₁)

/-- **Entanglement Gate Theorem**: J has interaction, hence any consistent combiner
    for J must be entangling (not separable). The RCL combiner satisfies this. -/
theorem entanglement_gate_theorem :
    NecessityGates.HasInteraction Cost.Jcost ∧
    IsEntangling Prcl ∧
    ¬ IsEntangling Padd := by
  refine ⟨NecessityGates.Jcost_hasInteraction, Prcl_entangling, Padd_not_entangling⟩

What this page does not claim

The converse, that zero mixed difference implies separability, is not proved. The theorem says nothing about quantum mechanical entanglement, only about the arithmetic of the combiner. The framework's cost function J is not claimed to be separable; in fact it is the entangling RCL combiner that is forced.

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

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