"The decay of hydrodynamic turbulence is a very fundamental problem. There have been no serious theoretical developments on the subject in the last seventy-five years or so. This paper presents a comprehensive and refreshing theory whose results are supposed to supersede most known results traditionally (K41, Heisenberg, and multifractal). If true, this is a major step in the field.

But is it true? The paper is based on a quantum ergodic hypothesis that stipulates an exact equivalence between the loop functional (the Fourier transform of the probability distribution) and the quantum trace of an evolution operator for the one-dimensional ring of Fermi particles. This appears plausible but there is no proof yet. But as the author points out, a plausible approach of theoretical physics does not wait for that proof.

The paper combines results from quantum field theory, concepts of classical hydrodynamics, number theory, etc., woven together in a very interesting way. The notion that classical power laws should be replaced by universal functions appears to be entirely correct. The comparison with experimental and DNS data (especially the length scale and the spectrum) is incomplete, as the author recognizes himself, partly because the experimental data are incomplete at present. The results of the paper may not be the final word on the problem, but it has so many interesting ideas in it that at least some of which will lead to a better understanding of the problem.

For that to happen, however, this paper has to be published first, and there is no better venue for it than this special issue. Thus, I strongly recommend the publication of the paper."

"Since the publication of this paper in Physics of Fluids, the basic equations (Euler ensemble as a solution of the Navier-Stokes loop equation) were mathematically verified at the Institute for Advanced Study (IAS). Additionally, new DNS simulations at a 4096³ lattice confirmed my predictions within the statistical margin of error. Recent physical experiments have also observed the energy decay index of 1.25, in complete agreement with my theoretical prediction.

What is left is a confirmation at very large-scale DNS, which I am awaiting to appear sometime this winter."

"The paper primarily focuses on the slope of energy spectra and its decay. The proposed approach captures the energy decays well, but the author proposes a steeper -7/2 spectra than K41. Kolmogorov spectra has been well documented in the literature, so this reviewer is very skeptical of the results.

Overall, this reviewer finds this paper too narrow in scope. Even if the mathematics is correct, and the proposed spectra is correct, this reviewer as a turbulence modeling and application engineer finds very little information that can be used to advance CFD."

"I appreciate the reviewer's extensive experience and dedication to understanding the physics of turbulence. Having spent over three decades on this problem myself, I deeply respect the historical role of traditional approaches, including the Kolmogorov -5/3 law (or the equivalent 2/3 law for the second moment of velocity difference). However, I believe it is essential to verify phenomenological assumptions with real and numerical experiments continually.

While older experiments for forced turbulence approximately confirmed this law, recent DNS studies over the past decade have shown small but significant deviations from K41 in forced turbulence. The multifractal model was developed to explain these deviations. My paper discusses these experiments in detail, particularly in section X.A (Discussion: The myths and reality of turbulent scaling laws). In decaying turbulence, these deviations are not minor—they are substantial. My theory explains these deviations within two standard errors of experimental data, without any fitting parameters. The lack of the 2/3 law is evident from the experimental data (see section VIII: Comparing our theory with DNS, Figs. 14 and 15). Recognized experts such as K.R. Sreenivasan and Gregory P. Bewley have reached similarly negative conclusions about K41 laws in decaying turbulence, which I cite in my paper.

Therefore, I must respectfully disagree with the reviewer on this aspect."

"With our turbulence theories so stagnant, maybe this crazy theory is what we need..."

"These green curves at your plots—what are they: some simulations of the Euler ensemble?"

"No, Peter, these are analytical computations using Mellin integrals with zeta functions. I did them with Mathematica using the Gauss-Kronrod method for oscillating integrals along an imaginary line. These functions in the energy spectrum being universal, they need to be computed only once, and I am sure that the number theorists will compute them with all the necessary precision and more."

"Dear Sasha, congratulations on the great progress. ... Best wishes Giorgio"

"Not even wrong! We told Migdal that his theory does not work, but he would not listen... "

"Congratulations on great success! I was thinking of circulation dynamics myself, --could not imagine it would be solvable."

"It was a great talk today, unlike your first ENS talk in 2023. Now we understand your solution of the loop equations...

"What is the origin of the discontinuity in the momentum loop appearing in the continuum loop equation?"

"The discontinuity arises from the singularity in the second functional derivative of the Stokes functional with respect to the loop's shape. The first functional derivative is proportional to the cross product of vorticity and the loop’s tangent vector $\vec C'(\theta)$. Upon varying this with respect to $\vec C'(\theta')$, we encounter a singular term proportional to $\delta(\theta-\theta')$. In momentum space, this translates to $\vec P(\theta) \times \vec P'(\theta') \delta(\theta-\theta')$. Integrating over $\theta'$ leads to the discontinuity: \[ \Delta \vec P(\theta) = \int_{\theta-0}^{\theta+0} d\theta'\,\vec P'(\theta'). \]"

"How did you guess the regular star polygon solution?"

"For about a year, I attempted to solve fixed-point equations numerically, treating them as iterative vector equations for polygon vertices in momentum space. The iterations around the loop never converged to a single periodic solution, as required. I then realized that periodicity had to be protected by symmetry on a circle. This insight guided me toward considering random walks constrained to circular symmetry, which naturally led to the discovery of random walks forming regular star polygons."

"Can we summarize our discussions by stating that my solution is verified?"

"Yes, we have proven a theorem confirming that your discrete solution satisfies the regularized Navier–Stokes equations at the level of statistical distributions. The remaining open question concerns the existence and nature of the circulation distribution in the limit of vanishing molecular scale."

"The UV divergences of loop functionals are well-known in gauge theory. Such divergences do not prevent their practical use in theoretical physics once proper renormalization procedures are applied."