Gapped Phases with Non-Invertible Symmetries: (1+1)d

1- The paper contains a detailed discussion of a novel aspect that spontaneous breaking of non-invertible symmetries can lead to physically distinguishable vacua.
2- The paper is very comprehensive and rather detailed in its discussion of examples, starting with relatively simple and well-known ones and then turning its attention to more complicated and also new ones that have not been discussed in the literature so far
3- At many places, the paper uses different perspectives and approaches to reach the same conclusions. This is instructive and provides additional confidence in the validity of the results.

1- Referring to the acceptance criteria: The paper does NOT contain a clear conclusion summarizing the results and offering perspectives for future work. One could argue though that the introduction itself serves as a suitable summary.
2- Beyond the introduction the authors are very sparse with references. For readers who are not entirely familiar with previous developments in this very dynamic area it is frequently very hard to judge whether a statement that is made is (i) well-established knowledge (not worth citing a reference [according to them]), (ii) a working hypothesis, or (iii) something that is going to be established later by the authors. One of many examples: "For invertible symmetries, i.e. when the symmetry is described by a finite (0-form) symmetry group G, the Euler terms can all still be tuned to zero" below equation (2.3), stated without explanation or reference.
3- Referring to point 2: What the authors regard as well-established knowledge might differ from what the reader regards as well-established knowledge, either because the reader lacks the detailed experience in the area or because they have different attitudes concerning the mathematical level of rigor of the statement. From this perspective it would be appropriate to be much more detailed about references that the authors have in mind in connection with certain statements and also be clearer about what is an assumption or working hypothesis and what will be shown subsequently.
4- Referring to point 3: The authors should imagine someone reading their paper in many years time, without having all the relevant background knowledge that active researchers in the field might currently have. In my opinion it should still be valuable then and be so as a stand-alone paper. For instance, the classification of fiber functors in (3.22) very clearly requires an explicit reference to where this (specific) result was obtained. Similarly, the statement "The irreducible topological boundary conditions of the SymTFT Z(S) are captured (modulo Euler terms) by Lagrangian algebras in the Drinfeld Center Z(S)" requires a reference. These are just two of many examples.

The paper uses the recent SymTFT approach to classify gapped phases of matter in 1+1D and provides a detailed discussion of the relevant vacua and generalized order parameters, from a general perspective and in many concrete examples. One novel aspect is the realization that the spontaneous breakdown of non-invertible symmetries may lead to the existence of physically distinguishable vacua. Also the discussion of some examples seems to be new, especially of the symmetries $TY(Z_N)$.

The paper provides a very clear introduction into the topic and a nice summary of the general strategy and the results obtained. I also regard it as very valuable to see how the general theory of "generalized charges" that was obtained in a different paper applies concretely in 1+1D, a setting many people from various different scientific communities have experience and interest in. Even though the results for some of the examples are not truly new I personally regard it as important to see how they can be obtained within the powerful and very general framework of SymTFT and generalized charges.

The methodology and reasoning is very sound and explained in appropriate detail even though the referee was not able to validate all individual statements and computations in view of the available time. Concrete strengths and weaknesses have been highlighted before.

In my opinion, the paper opens a new pathway in an existing/emerging direction, with clear potential for multipronged follow-up work, thus satisfying one of the expectations for publications in SciPost Physics. It also satisfies acceptance criteria 1-5, with some reservations concerning the appropriate referencing to the literature (see above). Criterion 6 is clearly violated as was pointed out before.

In view of the well-written summary of the strategy and the results in the introduction I would still recommend the manuscript for publication after a minor revision.

1- Please address points 2-4 mentioned in the weaknesses above by being more specific about the origin of specific statements throughout the bulk of the text, including the citation of appropriate literature
2- Explicit examples of statements that would benefit from clarification or references are (and this list should not be viewed as exhaustive)
* The fiber functor classification around (3.22)
* Why are the two statements around (3.40) equivalent?
* "one may obtain physically indistinguishable vacua even if non-invertible symmetries are spontaneously broken" just before Section 3.5 (this will be derived later and that should be stated explicitly)
* below (4.12), what is "trivial" about this G-symmetry?
* "A general fact of every 3d SymTFT Z(S) is that any arbitrary topological boundary condition of Z(S) can be obtained by gauging the S symmetry of BsymS . "
* The statements below (4.19)
* The statements in the second paragraph of Section 4.2
* The statements around (4.20)
* Equation (4.29)
3- There is a long history of "string operators" and "string order", specifically in 1+1D and in connection with the notion of hidden symmetry breaking, that is currently completely absent from the references. The relevant original paper (which to my knowledge coined the term) appears to be https://doi.org/10.1103/PhysRevB.40.4709 and is, in my opinion, the minimum which should be cited
4- In (3.36) and the subsequent discussion there is an index $i$ in the coefficients $n_i^a$ that I found rather confusing (as it does not appear on the left hand side of the equation) and whose presence should be motivated explicitly
5- There is an $\omega$ appearing in (5.16) that appears to refer back to Section 4 while in the meantime $\omega$ was used for the anomaly of $Vec_G^\omega$. I would suggest to state again explicitly what $\omega$ is in this formula.
6- Papers that I would recommend citing at appropriate places:
* https://doi.org/10.1103/PhysRevB.79.045316
7- There are a few typos that should be corrected
* Below eq (3.32) there is no "following" rectangle identity but rather a figure that should be referenced
* identity $\to$ identify above (5.41) and (5.44)
* Below (7.17) (and maybe more generally) the footnote should be placed after the punctuation
* Footnote 21 appears in an incomplete sentence and it lacks the desired reference
* Appendix C would benefit from checking again for grammar, word twists, etc.
8- Probably quite a few papers cited in the references have been published in the meantime and should be updated.