Anonymous Report 2 on 2023-11-19 (Invited Report)
Strengths

1- The work proposes a novel analysis of the landscape of the mixed p-spin model using a two-point Kac-Rice formula.
2- The equations are used to study marginal minima, showing that they are always isolated except at the threshold energy.
3- In general, the paper provides several insights into the landscape of the model. Despite it was not possible to connect these properties to dynamics yet, they may provide valuable insights for future investigations.
Weaknesses

1- Clarity.
Report

This paper presents several insights into an open problem. Despite the new insights did not allow the author to connect dynamics and landscape properties, they represent a significant contribution that I believe is worth publishing. Except for a few conceptual questions that I hope the author clarifies in a rebuttal, my main concern is clarity. At the moment the paper looks unpolished and hard to read.

*Comments/Questions*

On page 9, the author points out that there are solutions with complexity 0 that do not show an extensive barrier "in any situation". First, this "in any situation" is quite unclear. Does the author mean above and below the threshold energy? Does this solution exist even at high energy?
Can the author comment on what this solution could imply?

At the technical level, I am confused by one of the constraints imposed in eq.16, when \sigma_1 couples with all replicated s_a. I was expecting a sum of \sigma_b.s_a over a and b. This may represent a rotation applied to all replicas along a reference direction, which is probably what the author did, but there is no comment about that. In general, it would have helped to specify the constraints enforced instead of writing simply "Lagrange multipliers" before eq.16.

The author analyses the problem using the Franz-Parisi potential, however, this analysis does not seem to matter in the paper. We can read a comment at the end of Sec.3.1 but without actual implications. It should be either removed or expanded, at the moment it seems just without purpose.

In general, the paper lacks in clarity. It has several sentences that miss a conclusion, paragraphs that seem out of context, references missing, and the final sections unpolished. I will proceed in order

1. Introduction:
• "We see arrangements of barriers relative to each other, perhaps...". Why "perhaps"? Second, where is this analysis carried out? In the results section, the author analyses stable minima and marginal states, I don't know where to look. Adding a reference would have helped.
2. Model
• After eq.3, the author comments on the replica ansatz, but this is out of place. We are still introducing the model. It would be better to have it at the end of the section (where indeed the author comes back to the same concept) or remove it entirely.
• fig.1, add a caption under each figure saying what they are (oriented saddles, oriented minima, etc), it is much easier to read.
• fig.2, elaborate a bit more in the main text. This is introduced at the of the section without any comment.
3. Results
• fig.3, "the dot-dashed lines on both plots depict the trajectory of the solid line on the other plot", which one?
• fig.3, "In this case, the points lying nearest to the reference minimum are saddle with mu<mu, but with energies smaller than the threshold energy", so? What is the implication? This misses a conclusion.
• Sec.3.1, the author comments on the similarity with the pure model, without explaining what is similar. What should we expect on the p-spin? At least the relevant aspects. It would also be useful to plot a version of Fig.3 for the p-spin. It would make the discussion easier to follow.
• "the nearest neighbour points are always oriented saddles", where do I see this?
• the sentence "like in the pure models, the emergence [...]" is extremely hard to parse and the paragraph ends without a conclusion. What are the consequences?
• at page 9, the author talk about \Sigma_12 that however has not been defined yet.
4. Calculation
• this section starts without explaining what is the strategy to solve the problem. Explaining how the following subsection will contribute to the solution without entering into the details of the computation would be of great help.
• "This replica symmetry will be important later" how? Either we have an explanation following or it should be removed.
• at the end of a step it would be good to wrap everything up. For instance, sec.4.2 ends with "we do not include these details, which are standard" at least give a reference. Second, add the final result.
• "there is a desert where none are found" -> solutions are exponentially rare (or something else)

I would suggest a rewriting, especially the last sessions (4-6). I understand the intention of removing simple details, but they should be replaced by comments. The impression (which can be wrong but gives the idea) is of some working notes where simple steps have been removed, resulting in hard-to-follow computations. Finally, I would also recommend moving these sections to an appendix (after acknowledgement and funding).

    validity: high
    significance: good
    originality: high
    clarity: poor
    formatting: reasonable
    grammar: acceptable

Anonymous Report 1 on 2023-10-10 (Invited Report)
Strengths

1- In-depth replica-symmetric computation of the complexity of the mixed p-spin model, with special emphasis on the 3+4 model analysed by Folena et al.
2- Very well written with summary sections throughout.
Weaknesses

1- At times, the paper becomes overly technical. Some of the computations could be moved to the appendix to promote a more fluid and physically oriented reading.
2- Local stability analysis seems not to be enough to characterise and suitably distinguish the dynamical attractors.
Report

I believe that the paper meets the requirements of the journal and deserves to be published. I have included a list of suggested minor changes though.

i) On page 7, when referring to the set of marginal states that attract dynamics "as evidenced by power-law relaxations", it would be convenient to provide references for this statement.

ii) On the same page, the author refers to a quadratic pseudo-gap in the complexity function associated with marginal states. It would be helpful to have some more indication of how this was derived or, again, to provide appropriate references.

iii) Section 4 “Calculation of the two-point complexity”. The author states that conditioning the Hessian matrix of the stationary points to have a given energy and given stability properties influences the statistics of points only at the sub-leading order. It would be valuable to clarify the conditions under which this occurs. I was thus wondering whether the author can straightforwardly generalize such a computation and give some insights in the case of a sparse (no longer fully connected) model.

iv) Eq. (34) is quite complicated and difficult to grasp by eye. I thus wonder whether the numerical protocol is robust enough to be sure that by initializing differently, not exactly at q=0, the same solution is always found. How sensitive is the protocol to the choice of initial conditions?

v) In Section 5, the analysis of an isolated eigenvalue, which can be attributed to a low-rank perturbation in the Hessian matrix, is discussed. The technique results from a generalization of a paper recently published by H. Ikeda, restricted to a quadratic model though. It would be worthwhile to discuss how many of these predictions can be extended to models defined by a double-well potential or to optimization problems relying on non-quadratic functions (such as ReLu, sigmoid).
Requested changes

I found the paper interesting but quite technical in some points. Moving the saddle-point computations and part of the analysis (see for instance on pages 11-13 and 18-20) to the supplement would make it easier to capture the main results, especially for general readers without extensive expertise in the replica trick and these models.