Comment addressing report #2, fourth question

Discussed the interpretation of m given that the choice of x₀ is
arbitrary to the value of the Euler characteristic.

1 files changed, 1 insertions, 7 deletions

diff --git a/referee_response.md b/referee_response.md
index 08c463a..78518fe 100644
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@@ -8,12 +8,6 @@
 
 The manuscript is clear and self-contained. Below are some questions and requests for clarifications:
 
-(i) It is argue in the paper that in the regime V<V_sh, where the action at m=0 is complex, that in the regime where the action at, the solution should not be discarded as it indicates a negative average Euler characteristic. This argument is supported by the calculation of the second moment and the equality (18). Regarding the m* solutions, the action at m* becomes complex above V_on: is there an interpretation for these solutions in the regime V> V_on?
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-(ii) It is mentioned that the naive satisfiability threshold predicted from the vanishing of (12) coincides with the threshold obtained within the replica symmetric analysis of the cost function (3). By reading the manuscript, I have missed if/how the satisfiability threshold arising from the analysis of the average Euler characteristics compares with the threshold obtained from the zero-temperature analysis of the equilibrium problem with energy (3): could the Author comment on this?
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-(iii) Related to (ii): a general discussion on the instability of the average Euler characteristics to RSB is presented in Appendix C2, leading to the prediction (79). In the case of the spherical models illustrated in Figures 3 and 4, this instability seems not to be relevant in the regions where the SAT-UNSAT transition occurs. It is not clear to me whether all the cases illustrated in the plot are such that the correct ground state is found within a simple RS formalism from the T=0 equilibrium calculations, or whether there is no relation between the RSB instabilities occurring within the two calculations. A comment on this could be added in the text.
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 (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?
 Requested changes
 
@@ -33,7 +27,7 @@ Recommendation
  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. Maybe??
+ 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.
 
 # Report #3