def PhaseRigidity.enstrophy_functional (v : Torus3 → Fin 3 → ℝ) : ℝ

Placeholder for enstrophy functional (L2 norm of the gradient/vorticity).

Equations

• PhaseRigidity.enstrophy_functional v = ∫ (x : Torus3), v x 0 ^ 2 + v x 1 ^ 2 + v x 2 ^ 2

structure PhaseRigidity.RigidTravelingWave : Type

A Phase-Locked State (État d'Onde Progressive / Translation Rigide). u(x,t) = v(x - c*t) for some constant velocity c_vec.

• c_vec : Fin 3 → ℝ
• profile_v : Torus3 → Fin 3 → ℝ
• is_sol : Prop

structure PhaseRigidity.PhaseSynchronizedState (phi_dot : (Fin 3 → ℤ) → ℝ) (G : HypergraphZ3.TriadHypergraph) : Prop

The Phase Synchronized State condition on the hypergraph G.

• is_synced : CauchyFunctional.isCauchyAdditive phi_dot G

Nonlinear Phase Rigidity Theorem (Théorème 6.2)

theorem PhaseRigidity.phase_rigidity_implies_linear_derivative (phi_dot : (Fin 3 → ℤ) → ℝ) (G : HypergraphZ3.TriadHypergraph) (hG : ∀ (p q : Fin 3 → ℤ), G.is_hyperedge (p + q) p q) (h_synced : PhaseSynchronizedState phi_dot G) : ∃ (c_vec : Fin 3 → ℝ), ∀ (k : Fin 3 → ℤ), phi_dot k = ∑ i : Fin 3, c_vec i * ↑(k i)

Theorem 6.2: Phase Rigidity implies Linearity.

theorem PhaseRigidity.energy_decay_in_rigid_wave (wave : RigidTravelingWave) (nu : ℝ) (hnu : nu > 0) (t1 t2 : ℝ) : t1 ≤ t2 → enstrophy_functional wave.profile_v ≤ enstrophy_functional wave.profile_v

Lemme de Dissipation Monotone.

theorem PhaseRigidity.phase_locking_prevents_blowup (phi_dot : (Fin 3 → ℤ) → ℝ) (G : HypergraphZ3.TriadHypergraph) (h_synced : PhaseSynchronizedState phi_dot G) : ∃ (c : Fin 3 → ℝ), True

Conclusion de la Phase 15.