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 # Report #1
 
+## Response to referee comments
+
  * We fixed this typo.
  * The question of limits is a shrewd one, but ultimately the result is the same no matter how the calculation is done. Working directly at *M* = 1, the steps in the appendices are followed up to equation (28). With *M* = 1 and *V*₀² = *N**E*, the second term in the exponential remains of order *N* but the second is of order 1 and becomes another contribution to the prefactor. Comparing the resulting expression with (41) in the limit of α to zero with *V*₀² = *E*²/α, the two approaches result in the same effective action. In fact, an earlier version of this manuscript included two derivations, but the one for *M* of order 1 was deemed redundant in light of this. A note about this point has been added to the amended manuscript.
- * We agree, and further emphasized this in the amended manuscript.
+ * We agree, and further emphasized this by adding a sentence to the abstract of the amended manuscript.
 
 # Report #2
 
 ## Questions and requests for clarifications
 
- 1. In fact, the action is not complex when evaluated at m_* for V² > V_on² even though m_* itself becomes complex: the action remains real but becomes negative in this regime. This means that the contribution of these complex-m_* solutions in this regime shrinks with increasing N, and rather than representing a subleading but exponentially large (or even order 1) contribution to the Euler characteristic, their contribution is negligible.
- 2. The reference "A continuous constraint satisfaction problem for the rigidity transition in confluent tissues", which performs the FRSB treatment of the zero-temperature equilibrium problem for the case where f(q) = ½ q² and α = ¼, estimates V_SAT ≃ 1.871. Our calculation instead predicts V_SAT = 1.867229…. In private correspondence with the author of the quoted reference, they indicated that such a discrepancy could easily be due to inaccuracy in the numeric PDE treatment of the FRSB equilibrium problem and that they were not concerned by the seeming inconsistency. So, for the moment the two treatments are consistent but the agreement is not precise. A small discussion of this has been added in a footnote to the manuscript.
- 3. The irrelevance of RSB to the spherical spin glasses represented in the α → 0 limit of the included phase diagrams is expected. In both the pure spherical models (Fig. 3) and the mixed 1+2 models (Fig. 4) the equilibrium measure is always either replica symmetric or 1RSB, and the distribution of stationary points in both is always replica symmetric. However, the paper does include a discussion of the consistency between the RSB instability predicted by our second moment calculation and the appearance of RSB in the complexity of the spherical spin glasses, at the end of Appendix D (they are consistent). Not said in the initial manuscript is that this agreement also exists with the instability in the zero-temperature equilibrium measure, whose calculation is an intermediate step in finding the quenched shattering energy.
-   If the referee is also curious about the agreement between RSB instabilities in the zero-temperature equilibrium treatment of the cost function when α > 0, we addressed this briefly in the final paragraph of Appendix C. There are regions of the SAT–UNSAT transition for the case f(q) = ½ q² where the equilibrium cost function is FRSB, where this calculation does not have an instability. As noted in that paragraph, there are reasons to believe that this is a trait of the cost function itself, since the cost function is predicted to have such an instability for a mundane energy level set of the pure 2-spin spherical spin glass where no RSB occurs.
- 4. The picture described by the referee is partially true. The value of the Euler characteristic is independent of how x₀ is drawn, but this does not mean that elements of the calculation depending on m, the overlap with x₀, are unobservable. The simplest example is for the linear f(q) = q case, where for V₀² < V_SAT² the entire contribution to the Euler characteristic is made at m² > 0. The aspect that is malleable is at what value m_* the contribution is made. Since we draw x₀ uniformly on the sphere, m_* can be interpreted as the expected value of the overlap between a uniformly random point in configuration space and the nearest piece of the solution manifold. If x₀ were drawn in a different way, e.g., from a Boltzmann distribution on the cost function at finite temperature, then the value of the Euler characteristic computed would not change but the value of m_* would, and also our interpretation of its value. Whether such a change would modify the location of the onset transition V_on isn't known.
+ 1. In fact, the action is not complex when evaluated at *m*\_\* for *V*₀² > *V*\_on² even though *m*\_\* itself becomes complex: the action remains real but becomes negative in this regime. This means that the contribution of these complex-*m*\_\* solutions in this regime shrinks with increasing *N*, and rather than representing a subleading but exponentially large (or even order 1) contribution to the Euler characteristic, their contribution is negligible.
+ 2. The reference "A continuous constraint satisfaction problem for the rigidity transition in confluent tissues", which performs the FRSB treatment of the zero-temperature equilibrium problem for the case where *f*(*q*) = ½ *q*² and α = ¼, estimates *V*\_SAT ≃ 1.871. Our calculation instead predicts *V*\_SAT = 1.867229…. In private correspondence with the author of the quoted reference, they indicated that such a discrepancy could easily be due to inaccuracy in the numeric PDE treatment of the FRSB equilibrium problem and that they were not concerned by the seeming inconsistency. So, for the moment the two treatments are consistent but the agreement is not precise. A small discussion of this has been added in a footnote to the manuscript.
+ 3. The irrelevance of RSB to the spherical spin glasses represented in the α → 0 limit of the included phase diagrams is expected. In both the pure spherical models (Fig. 3) and the mixed 1+2 models (Fig. 4) the equilibrium measure is always either replica symmetric or 1RSB, and the distribution of stationary points in both is always replica symmetric. However, the paper does include a discussion of the consistency between the RSB instability predicted by our second moment calculation and the appearance of RSB in the complexity of the spherical spin glasses, at the end of Appendix D (they are consistent). Not said in the manuscript is that this agreement also exists with the instability in the zero-temperature equilibrium measure and the satisfiability threshold, whose calculation is an intermediate step in finding the quenched shattering energy.
+   If the referee is also curious about the agreement between RSB instabilities in the zero-temperature equilibrium treatment of the cost function when α > 0, we addressed this briefly in the final paragraph of Appendix C. There are regions of the SAT–UNSAT transition for the case *f*(*q*) = ½ *q*² where the equilibrium cost function is FRSB, where this calculation does not have an instability. As noted in that paragraph, there are reasons to believe that this is a trait of the cost function itself, since the cost function is predicted to have such an instability for a mundane energy level set of the pure 2-spin spherical spin glass where no RSB occurs (recall that for spherical spin glasses, the cost function is the square of the usual Hamiltonian).
+ 4. The picture described by the referee is partially true. The value of the Euler characteristic is independent of how *x*₀ is drawn, but this does not mean that elements of the calculation depending on *m*, the overlap with *x*₀, are unobservable. The simplest example is for the linear *f*(*q*) = *q* case, where for *V*₀² < *V*\_SAT² the entire contribution to the Euler characteristic is made at *m*² > 0. The aspect that is malleable is at what value *m*\_\* the contribution is made. Since we draw *x*₀ uniformly on the sphere, *m*\_\* can be interpreted as the expected value of the overlap between a uniformly random point in configuration space and the nearest piece of the solution manifold. If *x*₀ were drawn in a different way, e.g., from a Boltzmann distribution on the cost function at finite temperature, then the value of the Euler characteristic computed would not change but the value of *m*\_\* would, and also our interpretation of its value. Whether such a change would modify the location of the onset transition *V*\_on isn't known. A discussion of this issue has been added in a footnote to the amended manuscript.
 
 ## Requested changes
 
- 1. Ok.
- 2. Ok.
- 3. At the moment, when the manuscript is typeset Fig. 2 is on the same page as the description of the topological phases containing the requested information. Therefore, adding them to the caption feel redundant. However, if the referee feels strongly that the information should appear in both places the modification can be made.
- 4. Ok.
+ 1. We have done this with respect to point (ii), but not (iii) where discussion already existed in the manuscript.
+ 2. Some comments have been made.
+ 3. At the moment when the manuscript is typeset Fig. 2 is on the same page as the description of the topological phases containing the requested information. Therefore, adding them to the caption feels redundant. However, if the referee feels strongly that the information should appear in both places the modification can be made.
+ 4. Such a comment has been added.
 
 # Report #3
 
- 1. A discussion of the previous literature on this model has been added in the introduction.
- 2. A discussion of how to interpret the order parameter *m* has been added to the end of section 2.1.
+## Response to referee comments
+
+Since the weaknesses and the requested changes coincide, we address both simultaneously.
+
+ 1. The discussion of the previous literature on this model has been expanded in the introduction, as well as some small contextualization as to the content of our evidence of generic interest in the topic on the first page.
+ 2. A discussion of how to interpret the order parameter *m* has been added to a footnote in section 2.1.
  3. See the comments below.
-   * The referee is wrong to say that the Euler characteristic of a hypersphere is 2 independent of dimension. The Euler characteristic of all odd-dimensional manifolds is zero. Consider the cell complex on *S*₁ [pictured here](https://kent-dobias.com/files/S_1.png). The Euler characteristic calculated using the alternating sum over the number of cells of increasing dimension is χ(*S*₁) = 1 – 1 = 0.
-   * In this manuscript we present what we consider to be the simplest interpretation of the calculation, but the referee is correct to point out that a large Euler characteristic could indicate a complicated product manifold as well as one with many connected components, or other exotic manifolds besides. Our intuition for this is that applying one constraint amounts to taking a smooth, non-self-intersecting slice of a sphere, which should typically produce spheres of one fewer dimension. Repeating this reasoning recursively leads to the conclusion that the result is mostly unions of spheres all the way down. This schematic argument has been added to the manuscript as a footnote in section 2.3. As to what dynamics might look like in a problem where the manifold of solutions were actually a nontrivial product manifold, we have no idea.
- 4. The referee points out that previous work on gradient descent in the spherical spin glasses studied gradient descent from both uniformly random initial conditions ("infinite" temperature) and initial conditions drawn from a Boltzmann distribution at some finite temperature, and found that the final state of the dynamics reached marginal minima in a range of energies depending on the initial condition. The conjecture in this manuscript seeks only to explain the upper energy of this range, that associated with gradient descent from a uniformly random initial condition. Presumably there are a variety of behaviors observable by choosing initial conditions using a variety of initial distributions, Boltzmann or otherwise, and one day we may hope to address such questions using similar approaches to this paper. However, this is not addressed here. A small discussion of this point has been added to the manuscript.
-   * A paragraph addressing what might occur in planted models has been added to the manuscript.
+    * The referee is wrong to say that the Euler characteristic of a hypersphere is 2 independent of dimension. The Euler characteristic of all odd-dimensional manifolds is zero. Consider the cell complex on *S*₁ [pictured here](https://kent-dobias.com/files/S\_1.png). The Euler characteristic calculated using the alternating sum over the number of cells of increasing dimension is χ(*S*₁) = 1 – 1 = 0.
+    * In this manuscript we present what we consider to be the simplest interpretation of the calculation, but the referee is correct to point out that a large Euler characteristic could indicate a complicated product manifold as well as one with many connected components, or other exotic manifolds besides. Our intuition for favoring an interpretation with many connected components is that applying one constraint amounts to taking a smooth, non-self-intersecting slice of a sphere, and repeating this many times feels likely to lead to unions of mostly spheres. This schematic argument has been added to the manuscript as a footnote in section 2.3. As to what dynamics might look like in a problem where the manifold of solutions were actually a nontrivial product manifold or something more exotic, we have no idea.
+ 4. The referee points out that previous work on gradient descent in the spherical spin glasses studied gradient descent from both uniformly random initial conditions ("infinite" temperature) and initial conditions drawn from a Boltzmann distribution at some finite temperature, and found that the final state of the dynamics reached minima in a range of energies depending on the initial condition. The conjecture in this manuscript seeks only to explain the upper energy of this range, that associated with gradient descent from a uniformly random initial condition. Presumably there are a variety of behaviors observable by choosing initial conditions using a variety of initial distributions, Boltzmann or otherwise, and one day we may hope to address such questions using similar approaches to this paper. However, this is not addressed here. A small discussion of this point has been added to the manuscript.
+    * A paragraph addressing what might occur in planted models has been added to the manuscript.
  5. The existing citations to references regarding the use of superspace coordinates and operators have been clarified in the new manuscript, including an explicit reference to an explanatory appendix on the method written by the author. Repeating the same content here seems unnecessary. The relationship between the right and left-hand sides of (37–39) were made using the elementary rules outlined in the aforementioned appendix and symbolic algebra software, with no other intermediate steps to share.
-   * The subscript notation associated with the determinant has been explained in a footnote.
+    * The subscript notation associated with the determinant has been explained in a footnote.