Sequential Flows by Irrelevant Operators

I would like to thank the authors for their kind reply.
I believe they addressed the points I raised in a satisfactory way, with possibly the exception of point 1.

I still find the concept of a leading deformation in the context of this paper quite confusing.
The considered setup is obtained by applying two distinct and consecutive finite deformations and cannot be understood as being generated by a single irrelevant operator.
In fact, the deformed theory is controlled by three parameters, namely $\{λ_1,λ_2, λ_3\}$, and there is no point along the flow where these are all small. One can insist on studying such a regime, but it is probably just misleading. For instance, I don't see how the operator in Eq. (1.7) should be in any way related to the theory with finite $λ_3 = -λ_1 = -λ_2$, yet the authors state that "[...] the procedure of sequentially deforming that we described would seem to define some theory, whose leading order deformation is (1.7)."

1 - The authors should clarify the paragraph mentioned above.

I would like to thank the authors for their kind reply and clarifications in the paper.

Unfortunately, I believe that my remark about equations (3.22), (3.24) and (3.8) was misunderstood. The point was not to compare (3.8) to (3.22), but rather to explain why the authors obtained a different bound on $r$ from (3.24) than they did from (3.22). The reason is that to obtain (3.24), they used the same methodology as in (3.8), but I believe equation (3.8) is wrong.

To see this explicitly, one can check (3.8) in a special case, namely $\lambda_2 = \lambda_1$ and $\beta = \alpha$. The approximation (3.8) than reduces to $$\mathcal{E}_{n, m} \sim \frac{2 \sqrt{\alpha \lambda_1} - R}{\lambda_1 + \lambda_3} \ .$$ However, we can obtain the exact result for $\lambda_2 = \lambda_1$ and $\beta = \alpha$ by starting from (3.6): $$\mathcal{E}_{n, m} = \frac{R + \lambda_3 \mathcal{E}_{n, m}}{\lambda_1} \left( \sqrt{1 + \frac{4\lambda_1 \alpha_n}{(R + \lambda_3 \mathcal{E}_{n,m})^2}} - 1 \right) \ ,$$ which can be solved to give $$\mathcal{E}_{n, m} = \frac{-R + \sqrt{R^2 + 4 \alpha \lambda_1 + 8 \alpha \lambda_3}}{\lambda_1 + 2 \lambda_3} \ .$$ This result is exact and does not agree with the approximation from (3.8). In particular, the numerators between the prediction from (3.8) and the exact result differ with a factor of 2 in front of $\lambda_3$. I believe this is the same factor of 2 that distinguishes the bound $r < 1$ obtained in (3.24) from the actual bound $r < 1/2$ from (3.22).

What went wrong is that an implicit equation for $\mathcal{E}_{n,m}$ was used after the large $\alpha$ approximation (3.7), but $\mathcal{E}_{n,m}$ itself is of order $\sqrt{\alpha}$. Presumably one can obtain a correct version of (3.8) by taking into account the next order in (3.7).

Unless the authors disagree with this argument, I think it best to address this issue before proceeding with the publication of the paper.

Address the issue with (3.8) and (3.24).