Encyclopedia Foundation Foundation Dalembert Triangulated Proof

ARTICLE 5 claims 5 theorems

Foundation Dalembert Triangulated Proof

A machine-checked proof that one cost function is inevitable, by showing it passes four independent tests that its only rival fails.

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Four gates to one cost

The d'Alembert equation, H(t+u) + H(t-u) = 2H(t)H(u), is a classical functional equation whose smooth solutions are hyperbolic cosines. In this framework it appears as the fourth of four gates that together single out a unique cost function. The cost, a measure of how expensive a recognition event is, is the object under study. The four gates are: interaction, entanglement, curvature, and this d'Alembert structure. Each is a property a cost function may or may not have.

The framework's candidate cost, written J(x) = (x + 1/x)/2 - 1, passes all four gates. Its only rival in the classification, a flat quadratic cost, fails all four. The machine-checked library of formal theorems proves this consistency: J has interaction, its associated combiner is entangling, its log-lift satisfies the hyperbolic ODE G'' = G + 1, and it has d'Alembert structure. The flat cost has none of these properties. This is the triangulation: four independent characterizations all point to the same function.

The decisive step is the fourth gate. Given the structural axioms (normalization, symmetry, smoothness, calibration) plus d'Alembert structure, the theorem full_inevitability_four_gates forces F = J and then forces the combiner P to the form P(u,v) = 2uv + 2u + 2v. This completes the proof without any bridge hypothesis. The earlier three-gate version needed an unproved assumption that interaction forces the hyperbolic ODE; the four-gate version removes that assumption entirely. The d'Alembert identity is not an extra restriction but a derived signature of the hyperbolic cosine, and it closes the chain.

In Recognition Science, this matters because the cost function is the starting point of the whole forcing chain. Once J is fixed, the framework derives the golden ratio, an eight-tick recognition cycle, and three spatial dimensions. The triangulated proof is the load-bearing wall: it shows the cost could not be otherwise, not by fitting but by exclusion. The flat branch is not merely less elegant; it is ruled out by the same axioms that select J.

THEOREM gates_consistent · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean
gates_consistent · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean:190 
/-- **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⟩
THEOREM gates_consistent · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean
gates_consistent · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean:190 
/-- **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⟩
THEOREM full_inevitability_four_gates · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean
full_inevitability_four_gates · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean:217 
/-- **Full Inevitability with Four Gates**: d'Alembert structure completes the proof.

    Unlike the three-gate version which required a bridge hypothesis,
    the four-gate version is fully proved:

    d'Alembert structure + structural axioms ⟹ F = J ⟹ P = RCL
-/
theorem full_inevitability_four_gates (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (hNorm : F 1 = 0)
    (hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
    (hSmooth : ContDiff ℝ 2 F)
    (hCalib : deriv (deriv (fun t => F (Real.exp t))) 0 = 1)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
    (hDA : FourthGate.HasDAlembert F) :
    -- Part 1: F = J
    (∀ x : ℝ, 0 < x → F x = Cost.Jcost x) ∧
    -- Part 2: P = RCL on [0,∞)²
    (∀ u v : ℝ, 0 ≤ u → 0 ≤ v → P u v = 2 * u * v + 2 * u + 2 * v) := by
  constructor
  · -- Part 1: F = J from d'Alembert structure
    exact FourthGate.dAlembert_forces_Jcost F hNorm hSymm hSmooth hCalib hDA
  · -- Part 2: P = RCL from F = J
    have hFJ := FourthGate.dAlembert_forces_Jcost F hNorm hSymm hSmooth hCalib hDA
    exact P_forced_from_FJ F P hCons hFJ
THEOREM InteractionForcesHyperbolicODE · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean
InteractionForcesHyperbolicODE · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean:124 
/-- **Key Hypothesis**: Interaction + Structural Axioms forces the hyperbolic ODE.

    This is the central bridge connecting the gates. It says:
    If F has interaction, symmetry, normalization, smoothness, and consistency,
    then the log-lift G satisfies G'' = G + 1.

    This is NOT yet fully proved from first principles, but is strongly motivated by:
    1. The counterexample (no interaction) ⟹ flat ODE
    2. J (has interaction) ⟹ hyperbolic ODE
    3. Entanglement forces a specific functional form

    We state it as an explicit hypothesis to make the logical structure clear.
-/
def InteractionForcesHyperbolicODE : Prop :=
  ∀ (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ),
    F 1 = 0 →
    (∀ x : ℝ, 0 < x → F x = F x⁻¹) →
    ContDiff ℝ 2 F →
    deriv (deriv (fun t => F (Real.exp t))) 0 = 1 →
    (∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) →
    HasInteraction F →
    SatisfiesHyperbolicODE (fun t => F (Real.exp t))
THEOREM full_inevitability_four_gates · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean
full_inevitability_four_gates · IndisputableMonolith/Foundation/DAlembert/TriangulatedProof.lean:217 
/-- **Full Inevitability with Four Gates**: d'Alembert structure completes the proof.

    Unlike the three-gate version which required a bridge hypothesis,
    the four-gate version is fully proved:

    d'Alembert structure + structural axioms ⟹ F = J ⟹ P = RCL
-/
theorem full_inevitability_four_gates (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (hNorm : F 1 = 0)
    (hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
    (hSmooth : ContDiff ℝ 2 F)
    (hCalib : deriv (deriv (fun t => F (Real.exp t))) 0 = 1)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
    (hDA : FourthGate.HasDAlembert F) :
    -- Part 1: F = J
    (∀ x : ℝ, 0 < x → F x = Cost.Jcost x) ∧
    -- Part 2: P = RCL on [0,∞)²
    (∀ u v : ℝ, 0 ≤ u → 0 ≤ v → P u v = 2 * u * v + 2 * u + 2 * v) := by
  constructor
  · -- Part 1: F = J from d'Alembert structure
    exact FourthGate.dAlembert_forces_Jcost F hNorm hSymm hSmooth hCalib hDA
  · -- Part 2: P = RCL from F = J
    have hFJ := FourthGate.dAlembert_forces_Jcost F hNorm hSymm hSmooth hCalib hDA
    exact P_forced_from_FJ F P hCons hFJ

What this page does not claim

The bridge hypothesis InteractionForcesHyperbolicODE is not proved; it remains an explicit assumption in the three-gate version. This module does not prove the full forcing chain from J to three spatial dimensions; that chain lives elsewhere in the library. The d'Alembert gate is not an independent restriction in the Option A formulation; it is derived from the hyperbolic branch.

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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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