The manuscript by Wang et al deals with rigidity percolation in triangular networks. They introduce an anisotropic bond occupation probability and observe a rigidity transition in steps. Between the two transitions the Cxxxx modulus is nonzero while the other moduli are still zero. The topic is relevant to the readership of Soft Matter and the findings are definitely interesting. I am hesitant about the criterion of broad relevance and interdisciplinarity, as the authors do not spend much time discussion the implications of their findings for e.g. biological networks. Before deciding on my recommendation, however, there are some issues that need to be cleared up.

The authors are well-aware that networks near a rigidity threshold can exhibit highly nonlinear elastic behavior, as they cite the work by Sharma et al (Ref. 9 of the paper). Yet, for the purpose of the present study, they appear to only study stretching and not compression, and to fix the strain to a value of 10^(-3). The results are presented in a way that suggests the authors think that everything they do is in the linear regime, but it is not at all clear if this is the case. In fact, the inset of figure 1 which shows which bonds are load-bearing under stretching contains many structures that look like they would support tension but not compression, and might buckle under a compressive load. To address this question it would help to present a few stress-strain curves or modulus-strain curves for systems in the blue regime in figure 1. Or perhaps I am needlessly worried and the authors can provide some more details on how they choose the strains and how they verify linearity.

In connection to this, it is a pity that the authors appeared to have missed the work of Damavandi et al. (a pair of PREs in 2022) which sheds light on the work by Sharma and explains how one can think more generally about the difference between rigidity as it arises from the structure itself and rigidity as it appears as a consequence of deformation.

Overall the manuscript cites very few other papers and thus misses some that I would consider relevant, such as Feng and Sen (1984), Roux and Hansen (1987), Jacobs and Thorpe (1996).

A second issue is the misrepresentation of what Maxwell counting means in triangular lattices. Maxwell counting, done well, deals with counting _actual_ degrees of freedom and _independent_ constraints. It therefore does not predict that p_c should be 2/3 for rigidity percolation on the triangular lattice (as the authors claim in the conclusion section), because any system near p=2/3 will have some nodes that have no more connection to any other nodes, and therefore should no longer be included in counting the degrees of freedom, and it will have some constraints that are redundant, and therefore should not be included in counting the number of constraints. It has been pointed out in the past the the numbers reported in the literature for p_c (e.g. p_c=0.6602 by Jacobs and Thorpe) are only close to 2/3 by accident because the two effects described above nearly cancel each other. This issue arises in the conclusion section, but in a way also in the discussion surrounding the result for p_c reported here (0.645 ± 0.002) and its comparison to the result by Broedersz et al (0.651).

Finally a question out of curiosity: in the intermediate regime one expects C_xxxx>0 and the other moduli 0. In finite systems, of course the other moduli will no be strictly 0 for all samples. But this could happen in two ways: Either the xx direction is still a principal axis of the elasticity tensor but the other eigenvalues are nonzero due to finite size effects, or there is still a single axis along which the system is rigid but it happens to not be perfectly aligned with the x-axis. Do the authors have any idea which of the two interpretation is the more applicable one? (It is fine if the answer is no. Just curious.)

The authors have dealt with all comments in a very constructive manner and I can now recommend the article for publication in Soft Matter without any reservations.