Foundation Dalembert Triangulated Proof Interaction Forces Entanglement

The entanglement theorem

The declaration interaction_forces_entanglement is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states a precise conditional result: if a cost function F satisfies the framework's composition law, is normalized so that F(1)=0, is symmetric under reciprocals, and has interaction somewhere, then the combiner P that builds the cost of a product from the costs of its factors must be entangling. In plain words, the theorem proves that genuine interaction in the cost function forces the combining rule to have a cross term; the additive combiner, which merely sums two costs, is ruled out.

The framework's cost function measures the price of a recognition event, and the combiner is the rule that assembles the cost of a joint event from the costs of its parts. The theorem's assumptions are the framework's standard structural axioms: the composition law F(xy)+F(x/y)=P(Fx)(Fy), the normalization F(1)=0, and reciprocal symmetry F(x)=F(x⁻¹). The conclusion is that P is entangling, meaning its mixed second difference is nonzero. The proof is unconditional: it requires no bridge hypothesis and no additional regularity assumptions beyond those stated.

The theorem is one of four gates in the framework's triangulated proof structure. Gate 1 establishes that the canonical cost function J has interaction, while the flat counterexample Fquad does not. Gate 2, this theorem, shows interaction forces entanglement. Gate 3 characterizes the curvature of the log-lift, and Gate 4 adds the d'Alembert structure that completes the full inevitability theorem. The four gates are consistent: J passes all four, Fquad fails all four. The full four-gate theorem, proved in the same library, derives both F=J and P=RCL from the structural axioms plus the d'Alembert condition.

What the theorem does not claim is equally important. It does not, by itself, force the specific form of the cost function; that requires the additional d'Alembert gate. It does not claim that every combiner is entangling; the additive combiner is a proved counterexample. And it does not claim that interaction alone forces the hyperbolic ODE; that bridge remains an explicit hypothesis in the three-gate version, though the four-gate version completes the chain. The theorem is a sharp conditional: interaction plus the structural axioms yields entanglement, nothing more and nothing less.

/-- Interaction forces entanglement (under symmetry). -/
theorem interaction_forces_entanglement (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
    (hNorm : F 1 = 0)
    (hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
    (hInt : HasInteraction F) :
    IsEntangling P :=
  interaction_implies_entangling F P hCons hNorm hSymm hInt

/-- Additive combiner is not entangling. -/
theorem additive_not_entangling : ¬ IsEntangling Padd := Padd_not_entangling

/-- **Summary Theorem**: All four gates point to the same conclusion.

    - Gate 1 (Interaction): Distinguishes J from Fquad
    - Gate 2 (Entanglement): Characterizes RCL vs additive combiner
    - Gate 3 (Curvature): Characterizes hyperbolic vs flat ODE
    - Gate 4 (d'Alembert): Forces λ = 1 in cosh(λt), completing the chain

    All four gates are consistent: J passes all four, Fquad fails all four.
-/
theorem gates_consistent :
    -- J has all four properties
    HasInteraction Cost.Jcost ∧
    IsEntangling Prcl ∧
    SatisfiesHyperbolicODE Gcosh ∧
    FourthGate.HasDAlembert Cost.Jcost ∧
    -- Fquad/Padd have the opposite properties
    ¬ HasInteraction Counterexamples.Fquad ∧
    ¬ IsEntangling Padd ∧
    SatisfiesFlatODE Gquad ∧
    ¬ FourthGate.HasDAlembert Counterexamples.Fquad := by
  exact ⟨Jcost_hasInteraction, Prcl_entangling, Gcosh_satisfies_hyperbolic,
         FourthGate.Jcost_has_dAlembert_structure,
         Fquad_noInteraction, Padd_not_entangling, Gquad_satisfies_flat,
         FourthGate.Fquad_not_dAlembert_structure⟩

What this page does not claim

The theorem does not force the specific form of the cost function; that requires the d'Alembert gate. The theorem does not claim interaction alone forces the hyperbolic ODE; that bridge remains an explicit hypothesis in the three-gate version. The theorem does not apply to combiners outside the stated structural axioms.

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

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