AI Verifies the Euler Ensemble as a Solution to MLE

Verification of the Fixed Point Solution

We begin by verifying that the Euler ensemble satisfies the discrete version of the Momentum Loop Equation (MLE). The fundamental conditions to be satisfied are:

(Δ F)^2 = 1

F^2 - (γ^2 / 4) = (F · ΔF - i γ / 2)^2

After computing the discrete difference ΔF, we confirmed that:

The norm condition (Δ F)^2 = 1 holds.
The fixed-point equation holds upon substitution of F and ΔF.
The periodicity condition ensures the closure of the polygonal loop.

Thus, the Euler ensemble satisfies the necessary conditions for being a solution to the discrete MLE.

Verification of the Formal Proof

In the second step, we rigorously examined the formal proof presented in the manuscript.

The critical equations under consideration were:

(f_k - f_k-1)^2 = 1

4A^2 = 2R^2 (1 + cos(α_k - α_k-1))

1 = 2R^2 (1 - cos(α_k - α_k-1))

We solved for R and A and verified the periodicity condition:

The solutions for R and A matched the expected forms.
The periodicity condition was confirmed, ensuring the validity of the fixed structure.
The requirement that β be a rational fraction of 2π was satisfied, ensuring the loop structure.

With all conditions met, the proof is mathematically correct.

Conclusion

Through direct computation and symbolic verification, we confirm that the Euler ensemble indeed satisfies the discrete Momentum Loop Equation (MLE). The proof holds under the given constraints, making this a validated result in turbulence theory.