Two infinite families of facets of the holographic entropy cone

The paper demonstrates that two infinite families of holographic entropy inequalities constructed in ref.[13] are in fact facets of the holographic entropy cone (as opposed to redundant inequalities obtainable as a positive combination of other holographic inequalities). The proof method consists of explicit construction of a sufficiently large collection of star graphs (mostly perfect tensors, supplemented by "flower graphs") whose entropy vectors are linearly independent and saturate the inequalities. Although a vested reader might glean further insight into the inequalities from these derivations, the paper does not offer particularly illuminating lessons beyond a heuristic observation regarding distribution of entanglement. As such, I think the main value for the broader readership is the statement itself, i.e. the verification of facetness. This contributes a new result to the holographic entropy cone story. The paper meets the acceptance criteria and thus merits publication.

Here are optional suggestions regarding the presentation, in order of decreasing importance.

1- The introduction of facets could be made clearer.

(a) First, I would suggest specifying the characterization of inequalities as taking the form $\text{LHS}_i \ge \text{RHS}_i$ before eq.(1.1), and preferably before marginal consistency is specified. In other words, I would find it more logical to have (1.3) and (1.4) precede (1.1).

(b) Second, marginal consistency (1.1) should be explicitly supplemented with all other holographic entropy inequalities being satisfied (otherwise it would fail consistency in the first place); perhaps the authors also wish to further distinguish $\text{LHS}_j \ge \text{RHS}_j$ from $\text{LHS}_j > \text{RHS}_j$ for all $j\ne i$.
Indeed, the phrase "hyperplanes of marginal consistency" below (1.4) would (by conventional usage; see point 5) imply the latter, in which case I'd suggest further refinement of (1.1) replacing "for some $i$" with "for a single $i$."

(c) Finally, I think it's important to already indicate explicitly the key point that a requirement for a facet is not merely a true inequality, but also a non-redundant one. In other words, the characterization of (1.5) does not suffice without this additional characterization of the collection $\text{LHS}_i \ge \text{RHS}_i$. Otherwise, it may either not be possible to saturate the true inequality anywhere inside the cone (except for the origin), or it may only be possible to saturate it on a higher ($>1$) codimension locus. In particular, it might be pedagogically opportune after (1.5) to mention, in contrast to MMI, other inequalities (such as SSA) which are not facets.

2- The first bullet point on p.3, "They subsume many known facets of the holographic entropy cone, which were discovered in [8–11]." is rather vague. It would seem more natural to indicate how many, or the fraction of a specified number, and perhaps even which ones (or refer to the specification on p.5). Also, the reader may wonder whether or not the inequalities found in [12] are included in this statement, or why this work is not referenced among the list known inequalities here.

3- The statement on p.3 that "As all previously known facets are isolated (not part of any known infinite families)..." could be accompanied by a clarifying parenthetical remark regarding the dihedral family (e.g., up to how many parties was facetness established previously), to forestall the reader's potential confusion of why these are not a counter-example to the assertion.

4- The Lemmas are not always phrased in a form of a mathematical statement; some are accompanied by proofs, some by examples/comments, some by definitions. Adhering to a more conventional usage of Lemma might seem cleaner.

5- The usage of the word "hyperplane" is unconventional. Typically, a hyperplane is defined as a codimension-1 subspace, whereas the authors co-opt this for higher codimensions as well.

6- Just above Lemma 9 (p.10) in the sentence "Our next task ... codimension-2 hyperplane" it could be worth to re-specify "in the span of K4-vectors."

7- There are two occurrences of $\mathbb{RP}^2$ (on pp.17 and 18) which look unnecessarily technical -- one could either define this already as a name for (2.4), or just stick to "projective plane" throughout.

Publish (meets expectations and criteria for this Journal)

1) This paper proves an interesting result in full technical detail.
2) The paper makes good decisions about which portions of the proof to relegate to the appendices, and which lemmas are important to the understanding of the proof and thus need to be in the main text.
3) The paper nicely puts their result in the context of the larger program to better understand and describe the holographic entropy cone.
4) The proof is mathematically solid, and its method provides a path for a possible interpretation of the toric family of inequalities in terms of 4- and 6- party entaglement.
5) The discussion nicely relates to the physics content of the inequalities at issue; the uniformly-weighted star graphs corresponded to old black holes with each leg corresponding to a previously emitted Hawking quantum. This reader will be interested to see further understanding along these lines.

All weaknesses are minor and easily remediable as per the requested changes below.

This paper proves that two infinite families of holographic entropy inequalities are as tight as possible. That is, for each inequality, they construct $2^N-2$ linearly independent saturating holographic entropy vectors, showing that the achievable saturating surface is codimension one in the $2^N-1$ dimensional entropy space.

The result is of interest first because it completes the mathematical understanding of this set of inequalities (by understanding when they can be saturated). It improves the understanding of the holographic cone, as some previous facets now have a generalization as facets. And most importantly, its relation to understanding the nature of old black holes, as well as to entanglement in four and six party arrangements, is likely to produce interesting results in the future.

As the abstract admits, the proof itself is technical, but the paper does a good job of presenting its main ideas in the body while relegating more technical details to the appendix. The techniques used in the proof are also relevant; as pointed out in the conclusion, the fact that only star-graphs are needed, and the interplay between the $K4$ and $K6$ vectors, may point to an interpretation of these inequalities in terms of 4 and 6 party entanglement.

1) Apparent typo in right hand side of equation 3.2, one argument reads $X_0^1, X_1^1, X_2^2,X_4^1$ but should read $X_0^1,X_1^1,X_2^1,X_4^1$.
2) Figure 5 in the appendix should say that $g(1)$ is marked in red while $g(2)$ is marked in green.
3) Occasional unusual phrasings , e.g. "How to decide on that?" or "No because otherwise".

Publish (easily meets expectations and criteria for this Journal; among top 50%)