Foundation Dalembert Triangulated Proof Jcost Hyperbolic Ode

The hyperbolic signature

The hyperbolic cosine, written cosh, is the curve a hanging chain makes between two supports, and the same curve describes how a cost function must bend if it is to be the unique one forced by the Recognition Science framework. The declaration Jcost_hyperbolic_ODE states that the log-lift of the framework's cost function J satisfies the ordinary differential equation G'' = G + 1, where G is the function obtained by composing J with the exponential map. In plain terms, the second derivative of the log-lift equals the log-lift itself plus one, a hallmark of hyperbolic growth, not quadratic or linear behavior.

The classical background matters here. The equation G'' = G + 1 has two solution families: the hyperbolic branch G(t) = cosh(t) - 1, and the flat branch G(t) = t²/2. The framework's central cost function J(x) = (x + 1/x)/2 - 1, when lifted through the logarithm, lands exactly on the hyperbolic branch. The theorem Jcost_hyperbolic_ODE is the machine-checked statement that this is so, and it is paired with a companion result showing the flat candidate Fquad satisfies the competing equation G'' = 1 instead. The two equations are mutually exclusive, so the declaration pins down which side of the divide the framework's cost function occupies.

In Recognition Science, the framework models cost as a forced quantity: a discrete record of recognition events whose price is not chosen but derived. The declaration is one gate in a four-gate triangulated proof. The interaction gate distinguishes J from the flat counterexample, the entanglement gate characterizes the combiner P, the curvature gate is this hyperbolic ODE, and the d'Alembert gate supplies an alternate characterization. The four gates are consistent, and the theorem gates_consistent records that J passes all four while the flat candidate fails all four. The full inevitability theorem, which would force J from first principles without assuming the hyperbolic ODE, remains a stated hypothesis, not a proved result.

What the declaration does not claim is as important as what it proves. It does not assert that the hyperbolic ODE is forced by interaction and structural axioms alone; that bridge is explicitly marked as a hypothesis in the library. It does not claim that the d'Alembert identity is an additional restriction, since in the Option A formulation that identity is derived from the hyperbolic branch, not imposed. And it does not say anything about the empirical values of physical constants, which live in a separate part of the framework. The declaration is a precise, narrow statement: the log-lift of J satisfies G'' = G + 1, and that fact is machine-checked.

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

/-- Fquad's log-lift satisfies the flat ODE. -/
theorem Fquad_flat_ODE : SatisfiesFlatODE Gquad := Gquad_satisfies_flat

/-- The two ODEs are mutually exclusive. -/
theorem flat_not_hyperbolic : ¬ SatisfiesHyperbolicODE Gquad := Gquad_not_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 hyperbolic ODE is forced by interaction and structural axioms alone. The d'Alembert identity is an additional restriction beyond the hyperbolic branch. The declaration concerns any empirical physical constant.

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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