| Lossless discrete specializations | statement | corollary 10.2 | arxiv |
| Affine finite-type criterion | statement | theorem 12.6 | arxiv |
| Equal-marginal collapse | statement | proposition 12.1 | arxiv |
| Influence decoding | statement | lemma 12.5 | arxiv |
| Nested grouped roofs | statement | theorem 9.1 | arxiv |
| Nonnegative aggregation | statement | corollary 2.4 | arxiv |
| Exact negaform and compatible rooted certificate | definition | definition 5.1 | arxiv |
| Certified Gaussian inequality | statement | proposition 13.2 | arxiv |
| Explicit-input NP-hardness | statement | theorem 13.1 | arxiv |
| Degree-twelve certificate recovery | statement | corollary 5.6 | arxiv |
| Junction-tree formulation | statement | proposition 9.2 | arxiv |
| Transfer by rounding | statement | proposition 10.1 | arxiv |
| Finite-type invariance | statement | lemma 12.4 | arxiv |
| Class relations | statement | proposition 6.1 | arxiv |
| Recognition at fixed local arity | statement | proposition 6.2 | arxiv |
| Representation-optimized boundaries | statement | theorem 8.2 | arxiv |
| Strict degree-thirteen portfolio gap | statement | theorem 8.4 | arxiv |
| Exact cubic DR condition | statement | lemma 4.1 | arxiv |
| Arity-four recovery | statement | theorem 6.3 | arxiv |
| Cubic approximation | statement | theorem 4.5 | arxiv |
| A continuous roof is not necessarily a Boolean concave closure | example | example 2.2 | arxiv |
| Degree-twelve atomwise recovery | statement | theorem 5.5 | arxiv |
| Root independence for a fixed negaform | statement | proposition 5.2 | arxiv |
| A separation from the degree-twelve certificate cone | statement | corollary G.1 | arxiv |
| Box-constrained submodular quadratic maximization | statement | corollary 3.2 | arxiv |
| Outcome-space transfer | statement | proposition K.1 | arxiv |
| Multilinear lower-box vertices | statement | lemma 9.5 | arxiv |
| Optimal common retention game | statement | theorem 9.6 | arxiv |
| MKT4QM6FFF | statement | lemma E.1 | arxiv |
| Modular fractional-to-prophet guarantee | statement | corollary K.2 | arxiv |
| Fixed-roof joint frontiers | statement | theorem 8.1 | arxiv |
| Orientation theorem | statement | theorem 4.2 | arxiv |
| Explicit-input UGC-hardness | statement | theorem 13.3 | arxiv |
| Separate concavity | statement | lemma 2.1 | arxiv |
| Monotone bounded-arity recovery | statement | theorem 7.1 | arxiv |
| Mixed-polarity domination | statement | lemma 5.3 | arxiv |
| Monotone local objectives | statement | corollary 11.1 | arxiv |
| Finite map--adversary duality | statement | proposition 9.3 | arxiv |
| Correlation consequence | statement | proposition 10.3 | arxiv |
| Mixed component families | statement | corollary 6.5 | arxiv |
| A convex coordinate and a fractional optimum | example | example 3.3 | arxiv |
| Cubic certificate construction | statement | proposition 4.3 | arxiv |
| Upper bound and recovery | statement | theorem 2.3 | arxiv |
| Type-flow identity | statement | proposition 12.2 | arxiv |
| Cubic scalar recovery | statement | lemma 4.4 | arxiv |
| Quadratic half approximation | statement | theorem 3.1 | arxiv |
| Best-so-far guarantees | statement | proposition 9.7 | arxiv |
| Exact affine dual | statement | proposition 12.7 | arxiv |
| Exact cardinality closure | statement | lemma 7.2 | arxiv |
| Common-map dual certificate | statement | proposition 9.4 | arxiv |
| Binary maximal correlation | statement | lemma 12.3 | arxiv |
| Online preservation | statement | theorem 11.2 | arxiv |
| Arity-four optimization | statement | corollary 6.4 | arxiv |
| Optimal common-coordinatewise single-map factor | statement | theorem 5.4 | arxiv |
| Exact aggregate local-roof factor | statement | theorem 8.3 | arxiv |
| Sharp cardinality recovery | statement | theorem 7.3 | arxiv |
| Fixed-terminal equivalence | statement | proposition 3.4 | arxiv |
