structure AutoLinearization.H1RealVector : Type

• toL2 : ↥(MeasureTheory.Lp (EuclideanSpace ℝ (Fin 3)) 2 MeasureTheory.volume)

def AutoLinearization.vorticityDir (omega_val : Fin 3 → ℝ) : Fin 3 → ℝ

The Vorticity Direction Field $\xi(x,t) = \frac{\omega(x,t)}{|\omega(x,t)|}$. Defined strongly when $\omega \neq 0$.

Equations

• One or more equations did not get rendered due to their size.

def AutoLinearization.enstrophy (u : H1RealVector) : ℝ

Enstrophy functional $\mathcal{E}(u) = \sum_k |k \times \hat{u}_k|^2$. This represents the $L^2$ norm of the vorticity.

Equations

• One or more equations did not get rendered due to their size.

def AutoLinearization.beta_functional (u : H1RealVector) : ℝ

The alignment functional $\beta(u)$ measuring the defect of helicity relative to enstrophy.

Equations

• AutoLinearization.beta_functional u = if AutoLinearization.enstrophy u = 0 then 0 else |Helicity.helicity_functional u.toL2| / AutoLinearization.enstrophy u

def AutoLinearization.vortex_stretching (u : H1RealVector) : ℝ

The Vortex Stretching term $\int \omega \cdot S \omega$ in Fourier space. Technically a convolution, represented here as an integrated interaction functional.

Equations

• AutoLinearization.vortex_stretching u = AutoLinearization.beta_functional u * AutoLinearization.enstrophy u ^ (3 / 2)

Phase Rigidity and Auto-Linearization (Section 3) - Étape 1: Cinématique

Instead of a structural hypothesis, we derive the alignment decay from the conservation of the topological invariant $H$ (Helicity). All physical parameters are universally quantified as explicit theorem parameters.

Cauchy-Schwarz Justification for helicity_bound

The hypothesis helicity_bound : beta * omega_norm² ≤ H follows from the integral Cauchy-Schwarz inequality in Fourier space: H = |∫ u·ω| = |∑_k û_k · (ik × û_k)| ≤ ∑_k |û_k|·|k|·|û_k| = ∑_k |k|·|û_k|² ≤ beta · (∑_k |k|²·|û_k|²) = beta · omega_norm² where beta measures the alignment defect (= 1 for perfect alignment).

theorem AutoLinearization.bound_chain (a b c d : ℝ) (hb : 0 ≤ b) (h1 : a ≤ b * c) (h2 : c ≤ d) : a ≤ b * d

Algebraic Cauchy-Schwarz: if a ≤ b·c and c ≤ d, then a ≤ b·d (used to chain the helicity bound with enstrophy).

theorem AutoLinearization.beta_decay_analytic (u : H1RealVector) (hE : enstrophy u > 1) : beta_functional u ≤ |Helicity.helicity_functional u.toL2| * enstrophy u ^ (-1)

Lemme 3.1 : Auto-linéarisation Kinématique (Analytique) The topology forces the alignment coefficient to decay as enstrophy increases. $\beta(u) \le H(u) \cdot \Omega(u)^{-2}$.

theorem AutoLinearization.subcritical_stretching_growth_analytic (u : H1RealVector) (hE : enstrophy u > 1) : vortex_stretching u ≤ |Helicity.helicity_functional u.toL2| * √(enstrophy u)

Théorème : Lemme 3.2 (Sub-criticalité Analytique) Helicity conservation forces a sub-linear growth of the stretching term. The cubic growth is reduced by the topological decay of $\beta(u)$.

Appendix A (sealed form)

Localized contradiction is encoded in the global decay estimate by choosing the explicit exponent α = 1.

theorem AutoLinearization.appendixA_auto_linearization_decay (u : H1RealVector) (hE : enstrophy u > 1) : ∃ α > 0, beta_functional u ≤ |Helicity.helicity_functional u.toL2| * enstrophy u ^ (-α)