A Generalized Construction of Calabi-Yau Models and Mirror Symmetry

We have revised the article throughout, responding to all of the Referees' comments and questions (see below), and have used the opportunity to further clarify the narrative and update the bibliography.

To the Referee 1:

1) We have thoroughly rewritten Appendix A, expanding to spell out the motivations and details of the "working definition" that has been used throughout the analysis reported in this article. We also clarify (in several places including the conclusions) that we have verified, wherever possible, that the so-obtained results are both self-consistent, and also consistent with the by now well established results of Refs. [1,5,6].

2) In the text preceding Eq.(16), we have changed the specification of $\nu_\rho$ from "...vertices..." to "...the $N$-integral points $\nu_\rho$ of which being the minimal generators..." as this covers also the exceptional $m=2$ case, when $\nu_1=(-1,\dots,-1,0)$ is in a face, corresponding to the MPCP exceptional set.

3) The sentence containing Eq.(15) now correctly qualifies its use for reflexive (and convex) polytopes.

4) The definition is inserted in footnote 9, at the first mention of the "degree," on p.9. Its effective use in counting is mentioned right after Construction 3.1.

5) We have rewritten Construction 3.1 so as to refer to the full disjoint union of all faces, with the non-convex ones subdivided into convex ones. In our experience, this is very redundant, but does not introduce undue quandaries of non-uniqueness and their relevance. In turn, the subdivision of non-convex k-faces into convex k-faces poses no problems, since they are all k-coplanar so that their polars coincide. This has been clarified in the revised document.

6) The trans-polar polytope is assembled "from ground up," using the particular "dually" implied relations, which are now spelled out as Eqs. (17) at the end of Step 3 in Construction 3.1. Their use is illustrated in the explicit construction in Section 3.2, and we amplified the discussion of the observed sufficiency of the specifications in Construction 3.1 in the narrative on p.9-10.

7) As now clarified in several places, we do not have a proof that the specifications of Construction 3.1 always suffice, but have observed that they do in dozens upon dozens of examples (including some infinite sequences) purposefully invented to test it. Indeed, we wholeheartedly welcome any and all attempt to either prove or improve Construction 3.1 --- and in a twin fashion also provide a conclusive definition of VEX polytopes as a maximal closure of trans-polar pairs of polytopes.

8) We have included an explicit reference to star-domains, noting however that they must be understood in a generalized fashion, implicit in the (cited) literature on "multi-fans."

To Referee 2:

1. We have thoroughly rewritten Appendix A, expanding to spell out the motivations and details of the "working definition" that has been used throughout the analysis reported in this article. We also clarify (in several places including the conclusions) that we have verified, wherever possible, that the so-obtained results are both self-consistent, and also consistent with the by now well established results of Refs. [1,5,6].

2. In the rewritten Appendix A (as well as in Section 2), we highlight the places where and why the related notions of "base-locus" and "transversality" (and Batyrev's $\Delta$-regularity) turn up. We now also explicitly state that our article addresses the low-energy regime of the considered GLSMs, and defer the UV analysis.

3. The related issues of "intrinsic limits," transversality and non-singularity are detailed in Appendix A, and noted also elsewhere throughout this revision.