Changes and comments addressing report #2

 - Indicated which of the requested changes were made.
 - Modified the manuscript to indicated that the function H must be
   Morse, i.e., not have any degenerate stationary points.

2 files changed, 8 insertions, 5 deletions

diff --git a/referee_response.md b/referee_response.md
index 78518fe..d1f1f44 100644
--- a/referee_response.md
+++ b/referee_response.md
@@ -21,14 +21,18 @@ Requested changes
 Recommendation
 
 
-(iv) If understand correctly, the vector x0 is arbitrary and it is introduced with the purpose of decomposing the contributions to the Euler characteristics in terms of m. Given the arbitrarily of x0, one would naively expect that the “observable” part of the solution space corresponds to m=0, and that any analysis of the constraint satisfaction problem that is x0-independent should be unable to pick up the transition between Regime II and Regime III: is this the case?
-
  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.
 
+## 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.
+
 # Report #3
 
 Strengths
diff --git a/topology.tex b/topology.tex
index d3d90d4..56af0c6 100644
--- a/topology.tex
+++ b/topology.tex
@@ -247,9 +247,8 @@ points, one must take pains to eliminate the sign of the determinant
 \cite{Fyodorov_2004_Complexity}. Here it is correct to preserve it.
 
 We need to choose a function $H$ for our calculation. Because $\chi$ is
-a topological invariant, any choice will work so long as it does not share some
-symmetry with the underlying manifold, i.e., that $H$ satisfies the Smale condition. Because our manifold of random
-constraints has no symmetries, we can take a simple height function $H(\mathbf
+a topological invariant, any choice will work so long as it does not have degenerate stationary points on the manifold, i.e., that it is a Morse function, and does not share some
+symmetry with the underlying manifold, i.e., that it satisfies the Smale condition. Because our manifold is random and has no symmetries, we can take a simple height function $H(\mathbf
 x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with
 $\|\mathbf x_0\|^2=N$. We call $H$ a height function because when $\mathbf x_0$ is
 interpreted as the polar axis of a spherical coordinate system, $H$ gives the height on the sphere relative to the equator.