hyperbolic_not_flat

formal source

 117/-- The two ODEs are mutually exclusive. -/
 118theorem flat_not_hyperbolic : ¬ SatisfiesHyperbolicODE Gquad := Gquad_not_hyperbolic
 119
 120theorem hyperbolic_not_flat : ¬ SatisfiesFlatODE Gcosh := Gcosh_not_flat
 121
 122/-! ## The Bridge: Connecting the Gates -/
 123
 124/-- **Key Hypothesis**: Interaction + Structural Axioms forces the hyperbolic ODE.
 125
 126    This is the central bridge connecting the gates. It says:
 127    If F has interaction, symmetry, normalization, smoothness, and consistency,
 128    then the log-lift G satisfies G'' = G + 1.
 129
 130    This is NOT yet fully proved from first principles, but is strongly motivated by:
 131    1. The counterexample (no interaction) ⟹ flat ODE
 132    2. J (has interaction) ⟹ hyperbolic ODE
 133    3. Entanglement forces a specific functional form
 134
 135    We state it as an explicit hypothesis to make the logical structure clear.
 136-/
 137def InteractionForcesHyperbolicODE : Prop :=
 138  ∀ (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ),
 139    F 1 = 0 →
 140    (∀ x : ℝ, 0 < x → F x = F x⁻¹) →
 141    ContDiff ℝ 2 F →
 142    deriv (deriv (fun t => F (Real.exp t))) 0 = 1 →
 143    (∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) →
 144    HasInteraction F →
 145    SatisfiesHyperbolicODE (fun t => F (Real.exp t))
 146
 147/-! ## The Main Inevitability Theorem -/
 148
 149/-- **Full Inevitability Theorem (Triangulated Form)**
 150