gates_consistent

formal source

 196
 197    All four gates are consistent: J passes all four, Fquad fails all four.
 198-/
 199theorem gates_consistent :
 200    -- J has all four properties
 201    HasInteraction Cost.Jcost ∧
 202    IsEntangling Prcl ∧
 203    SatisfiesHyperbolicODE Gcosh ∧
 204    FourthGate.HasDAlembert Cost.Jcost ∧
 205    -- Fquad/Padd have the opposite properties
 206    ¬ HasInteraction Counterexamples.Fquad ∧
 207    ¬ IsEntangling Padd ∧
 208    SatisfiesFlatODE Gquad ∧
 209    ¬ FourthGate.HasDAlembert Counterexamples.Fquad := by
 210  exact ⟨Jcost_hasInteraction, Prcl_entangling, Gcosh_satisfies_hyperbolic,
 211         FourthGate.Jcost_has_dAlembert_structure,
 212         Fquad_noInteraction, Padd_not_entangling, Gquad_satisfies_flat,
 213         FourthGate.Fquad_not_dAlembert_structure⟩
 214
 215/-! ## Complete Inevitability with Four Gates -/
 216
 217/-- **Full Inevitability with Four Gates**: d'Alembert structure completes the proof.
 218
 219    Unlike the three-gate version which required a bridge hypothesis,
 220    the four-gate version is fully proved:
 221
 222    d'Alembert structure + structural axioms ⟹ F = J ⟹ P = RCL
 223-/
 224theorem full_inevitability_four_gates (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
 225    (hNorm : F 1 = 0)
 226    (hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
 227    (hSmooth : ContDiff ℝ 2 F)
 228    (hCalib : deriv (deriv (fun t => F (Real.exp t))) 0 = 1)
 229    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))