| Cutoff profiles at boundary atoms | statement | proposition 4.8 | arxiv |
| Consistency of finite-support approximation | statement | theorem 5.1 | arxiv |
| Mean-preserving scalar shadow | statement | proposition 3.4 | arxiv |
| Pointwise, local \(L^p\), and uniform profile limits | statement | corollary 4.9 | arxiv |
| Continuous convergence at atom-free cutoffs | statement | theorem 4.7 | arxiv |
| Optimal cuts, actions, and encodings | definition | definition 2.8 | arxiv |
| Half-line reduction (recalled | statement | proposition 2.6 | arxiv |
| Objective-gap and minimizer-retention certificate | statement | proposition 6.2 | arxiv |
| Retained axioms and state order | statement | proposition 3.1 | arxiv |
| Ordered and cutoff-monotone levels | statement | corollary 2.10 | arxiv |
| Order and additivity on a fixed partition | statement | proposition 3.3 | arxiv |
| Epi-convergence | definition | definition 4.10 | arxiv |
| Equivalent descriptions of a genuine one-jump limit | statement | proposition 4.14 | arxiv |
| Cutoff profiles | definition | definition 2.4 | arxiv |
| Outer stability of exact and approximate cutoffs | statement | corollary 4.12 | arxiv |
| Strong consistency of empirical regimes | statement | theorem 5.3 | arxiv |
| Subadditivity fails, even comonotonically | statement | proposition 3.7 | arxiv |
| Continuity of \(G\)-centering | statement | lemma 4.5 | arxiv |
| \(\psi\)-weak convergence | definition | definition 4.2 | arxiv |
| Two-regime scan with conditional error control | definition | definition 6.1 | arxiv |
| Moving restrictions | statement | lemma 4.6 | arxiv |
| Translation and positive-scale equivariance | statement | proposition 2.11 | arxiv |
| Finite-support prefix characterization | statement | proposition 6.1 | arxiv |
| Four-point specialization | statement | proposition 7.1 | arxiv |
| Useful characterizations | statement | lemma 4.3 | arxiv |
| Conditional centers | statement | proposition 2.5 | arxiv |
| Wasserstein approximation | statement | corollary 5.2 | arxiv |
| Global monotonicity fails | statement | proposition 3.6 | arxiv |
| Additivity along an affine ray | statement | corollary 3.2 | arxiv |
| Existence and interval structure | statement | proposition 2.2 | arxiv |
| Equivalence with two-state approximation | statement | corollary 2.7 | arxiv |
| Joint convergence of cutoffs, levels, and step graphs | statement | theorem 4.13 | arxiv |
| Cutoff-profile epi-convergence with boundary atoms | statement | theorem 4.11 | arxiv |
| Stochastic monotonicity of centers | statement | proposition 2.9 | arxiv |
| A vanishing tail with persistent loss | example | example 4.1 | arxiv |
| One-regime aggregation envelope | statement | proposition 3.8 | arxiv |
