| Gaussian comparison with quadratic confinement | statement | lemma 3.5 | arxiv |
| Dependence on the path | statement | lemma 4.1 | arxiv |
| Choice of the parameters | statement | lemma 4.6 | arxiv |
| Free energy, marks along the sampled leaf, and variance | statement | corollary A.2 | arxiv |
| Initial condition | statement | proposition 2.11 | arxiv |
| Upper bound at a critical path | statement | proposition 2.12 | arxiv |
| One-dimensional semi-concavity estimate | statement | lemma 4.3 | arxiv |
| Hamilton--Jacobi lower bound | statement | proposition 5.5 | arxiv |
| Adding one coordinate | statement | lemma 6.2 | arxiv |
| Gaussian comparison in covariance norm | statement | lemma 3.4 | arxiv |
| First-order radial expansion | statement | lemma 3.8 | arxiv |
| Bulk and true free energies | statement | lemma 3.11 | arxiv |
| Balanced models | definition | definition 7.1 | arxiv |
| Replacement of the radial perturbation | statement | lemma 3.9 | arxiv |
| Viscosity solution and Hopf formula | statement | theorem 5.4 | arxiv |
| Differentiability of $\bar F_N$ | statement | proposition 2.1 | arxiv |
| Gaussian block with one coordinate | statement | lemma 2.7 | arxiv |
| Multi-species initial condition | statement | corollary 2.10 | arxiv |
| Covariance of radial perturbations | statement | lemma 3.2 | arxiv |
| Viscosity solutions | definition | definition 5.3 | arxiv |
| One-species formula for balanced models | statement | corollary 7.3 | arxiv |
| Convexity of the initial condition | statement | lemma 2.9 | arxiv |
| Disintegration of the surface measure | statement | lemma 3.3 | arxiv |
| Invariance of tilted cascades | statement | lemma A.1 | arxiv |
| The inner infimum at a critical path | statement | lemma 2.13 | arxiv |
| Spherical cavity limit | statement | theorem 3.24 | arxiv |
| Removal of the cutoff | statement | lemma 3.23 | arxiv |
| One-dimensional bounds | statement | lemma 4.2 | arxiv |
| Ghirlanda--Guerra identities on average | statement | lemma 4.5 | arxiv |
| Semi-concavity of the free energy | statement | proposition 2.2 | arxiv |
| Boundedness of the cavity increment | statement | lemma 3.6 | arxiv |
| Monotonicity for the dual-cone order | statement | lemma 5.1 | arxiv |
| The multiplier equation | statement | lemma 2.4 | arxiv |
| Variance of the logarithm of a Poisson--Dirichlet sum | statement | lemma A.3 | arxiv |
| Radial self-averaging | statement | lemma 3.14 | arxiv |
| Continuity in the species proportions | statement | lemma 6.5 | arxiv |
| Divergence for non-admissible $b$ | statement | lemma 3.20 | arxiv |
| Replacement of the radial perturbation at fixed cutoff | statement | proposition 3.10 | arxiv |
| Quadratic gain from the cascade | statement | lemma 3.19 | arxiv |
| Uniform moments of the cavity coordinates | statement | lemma 3.22 | arxiv |
| Regularization | definition | definition 5.2 | arxiv |
| PMF04GH86J | statement | theorem 1.1 | arxiv |
| Fixed-cutoff chart-to-Gaussian comparison | statement | proposition 3.15 | arxiv |
| Expansion of the Hamiltonian in the chart | statement | lemma 3.7 | arxiv |
| Derivative of the initial condition | statement | lemma 3.18 | arxiv |
| Derivative matching | statement | lemma 4.4 | arxiv |
| Normalized form of the Hopf formula | statement | lemma 6.4 | arxiv |
| Regularity of the initial condition | statement | proposition 2.8 | arxiv |
| Finite-overlap approximation | statement | lemma 3.21 | arxiv |
| Moment bounds under the cavity tilts | statement | lemma 3.13 | arxiv |
| Identification of the limit for rational proportions | statement | corollary 6.1 | arxiv |
| Derivatives of random convex functions | statement | lemma 3.12 | arxiv |
| Finite-size comparison | statement | lemma 6.3 | arxiv |
| Balanced reduction | statement | proposition 7.2 | arxiv |
| Concentration of the free energy | statement | lemma B.1 | arxiv |
| Diagonal compensation at fixed cutoff | statement | lemma 3.16 | arxiv |
| Stationarity in $b$ | statement | lemma 3.17 | arxiv |
| Linear tilts of the cascade | statement | lemma 2.6 | arxiv |
| Regularity of the multiplier and the derivative path | statement | lemma 2.5 | arxiv |
| Tilts with a product structure | statement | lemma A.4 | arxiv |
