Foundation Dalembert Triangulated Proof Jcost Is Hyperbolic

The hyperbolic signature

The declaration Jcost_is_hyperbolic is a theorem in the machine-checked library of formal theorems that powers Recognition Science. It states that the cost function J, the object at the center of the framework, has interaction. In plain terms, the cost of recognizing two events together is not simply the sum of their individual costs; the pair carries extra structure that a simple additive account misses. The theorem's full name is Jcost_is_hyperbolic, and it belongs to a family of results that classify possible cost functions into distinct branches.

The classification is a trichotomy. One branch is flat, where the cost function is quadratic and the combining rule is additive. Another is spherical, and the third is hyperbolic. The theorem proves J lands in the hyperbolic branch, where the cost function behaves like the hyperbolic cosine minus one. This is not a definitional choice; it is a proved consequence of J's defining properties. The proof rests on a chain of lemmas that show J satisfies the hyperbolic ordinary differential equation G'' = G + 1, where G is the log-lift of J. That equation is the signature of the hyperbolic branch, and it is what the declaration establishes.

In Recognition Science, the result matters because it rules out the flat alternative. If J were flat, the combining rule would be additive and the framework would lose its distinctive structure. The theorem shows that J is not flat, and it also shows that the additive combiner is not entangling, while the hyperbolic branch's combiner is. The gates are consistent: J passes all four tests that characterize the hyperbolic branch, and the flat counterexample fails all four. The declaration is one gate in that larger consistency proof.

The theorem does not claim that interaction alone forces the hyperbolic branch. That stronger bridge statement, that interaction plus structural axioms implies the hyperbolic ODE, is explicitly left as a hypothesis in the library, not proved. The declaration only establishes that J, the specific cost function, is in the hyperbolic branch. It does not derive the full inevitability theorem, which requires the d'Alembert gate and additional assumptions. It also does not say anything about the spherical branch or about which cost functions are possible beyond the classification itself.

/-- J is in the hyperbolic branch (has interaction). -/
theorem Jcost_is_hyperbolic : HasInteraction Cost.Jcost := Jcost_hasInteraction

/-- J's log-lift satisfies the hyperbolic ODE. -/
theorem Jcost_hyperbolic_ODE : SatisfiesHyperbolicODE Gcosh := Gcosh_satisfies_hyperbolic

/-- **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 prove that interaction alone forces the hyperbolic branch; that bridge statement remains a hypothesis. The theorem does not derive the full inevitability theorem, which requires the d'Alembert gate and additional assumptions. The theorem says nothing about the spherical branch or about which cost functions are possible beyond the classification itself.

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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