| Legendre duality in the magnetization | statement | lemma 6.7 | arxiv |
| Convergence estimate | statement | proposition 8.5 | arxiv |
| Gibbs averages and response along the branch | statement | lemma 3.1 | arxiv |
| The selected leaf of a finite-level cascade | statement | lemma A.3 | arxiv |
| Factorization at $t=0$ | statement | lemma 7.2 | arxiv |
| Bounded gain in prediction | statement | lemma 3.2 | arxiv |
| Diffusion comparisons | statement | lemma 6.9 | arxiv |
| Two leaves selected from a finite-level cascade | statement | lemma A.4 | arxiv |
| Well-posedness, comparison, and Hopf formula | statement | theorem 8.1 | arxiv |
| Derivative identities | statement | theorem 4.1 | arxiv |
| Enriched free energy on one branch | definition | definition 2.4 | arxiv |
| Convergence of the frozen problem | statement | lemma A.2 | arxiv |
| It\^o's formula along the branch | statement | lemma 2.5 | arxiv |
| Enriched free energy with external fields | statement | theorem 10.1 | arxiv |
| Quantitative remainder estimate on a slice | statement | theorem 5.1 | arxiv |
| Barriers for the magnetization | statement | lemma 6.4 | arxiv |
| Limit free energy on a slice | statement | theorem 1.2 | arxiv |
| Well-posedness of the branch equation | statement | lemma 2.1 | arxiv |
| Two replicas at level $r$ as conditioned cascade replicas | statement | proposition A.6 | arxiv |
| Scalar martingale formula | statement | proposition 6.6 | arxiv |
| A block with zero path | statement | lemma 7.3 | arxiv |
| Localization | statement | lemma 9.3 | arxiv |
| Cancellation of one Hessian | statement | lemma 5.5 | arxiv |
| Lipschitz bound and extension | statement | proposition 4.2 | arxiv |
| Scalar dual identities | statement | lemma 9.5 | arxiv |
| Identification of the limit free energy | statement | theorem 1.1 | arxiv |
| Convexity at fixed magnetization | statement | proposition 6.10 | arxiv |
| Approximate solution in finite dimensions | statement | proposition 8.4 | arxiv |
| Cole--Hopf recursion | statement | lemma A.1 | arxiv |
| The slice initial condition | statement | corollary 6.11 | arxiv |
| The interpolation and its adjoint | statement | lemma 8.3 | arxiv |
| Entropy-corrected continuity in the magnetization | statement | lemma 7.4 | arxiv |
| The Hopf value through potentials | statement | proposition 9.6 | arxiv |
| Well-posedness and regularity | statement | lemma 6.1 | arxiv |
| Gluing the slices | statement | lemma 7.1 | arxiv |
| The branch as the limit of the selected leaves | statement | proposition A.5 | arxiv |
| Joint density | statement | lemma 5.2 | arxiv |
| Initial data on slices | statement | corollary 7.6 | arxiv |
| Symmetries | statement | lemma 2.7 | arxiv |
| Dual representation of $g_m$ | statement | lemma 9.4 | arxiv |
| Agreement with the cascade | statement | proposition A.8 | arxiv |
| Conjugate form of the Hopf formula | statement | lemma 9.2 | arxiv |
| Concentration given the sampled configuration | statement | lemma 5.6 | arxiv |
| Dictionary between the two parametrizations | statement | lemma 6.5 | arxiv |
| Comparison principle | statement | lemma 2.3 | arxiv |
| Comparison with error terms | statement | lemma 8.2 | arxiv |
| Initial data on one-species slices | statement | proposition 7.5 | arxiv |
| Representation by potentials | statement | theorem 10.2 | arxiv |
| Convexity in the Gaussian data | statement | lemma 5.3 | arxiv |
| It\^o identities along the field diffusion | statement | lemma 6.2 | arxiv |
| Transfer and attainment | statement | proposition 9.7 | arxiv |
| Concavity in the law and supporting functions | statement | corollary 6.8 | arxiv |
| The value at $q=0$ | statement | lemma 4.3 | arxiv |
| Variance of level averages | statement | proposition 5.4 | arxiv |
| Path biconjugacy | statement | lemma 9.1 | arxiv |
| Continuity in the law | statement | lemma 6.3 | arxiv |
