Replace raw neural scores with randomized pairwise-error probabilities relative to a reference candidate distribution. Use a fixed PEP threshold to accept, abstain, or form a variable-size candidate list; exact uniformity under the reference law makes the threshold interpretable independently of the model's score scale and robust to ties.
Constrain an embedding table to the unit sphere and penalize repeated or nearly repeated pair sums. This discourages additive quadruples a+b approximately equal to c+d, reducing unwanted linear structure and making distinct tokens less interchangeable under downstream composition. The theorem provides a geometric target: on a sphere, the affine-line concentration factor is bounded by two, so exact additive energy should scale close to n squared rather than the much larger values produced by…
Encode each scalar or discrete code t by the strictly convex lift γ(t)=(t,t²), optionally followed by a learned linear projection and normalization. Because three-code sums on this curve have only near-minimal additive energy, the representation should produce fewer collisions when a model composes three tokens, codes, or retrieved items by addition.
Replace one-step greedy landmark selection in Nyström attention or kernel compression with a restricted pairwise-lookahead rule. The lookahead is motivated by the paper's explicit obstruction: a signed triangle can make individual column gains exhibit increasing rather than diminishing returns, so the best next column need not belong to the best pair.
Augment a representation-learning objective with penalties enforcing the paper's four-point metric inequalities, and use an exponential snowflake kernel instead of unconstrained dot-product similarity. The experiment tests whether geometrically valid similarities improve retrieval or attention stability at equal model size and compute.
Replace one-hot node IDs or large positional encodings in a GNN with coordinates from a compact abelian Cayley graph. The coordinates preserve graph-shortest-path geometry exactly, while Fourier characters of cyclic factors provide smooth neural features with fewer channels.
Build an attention or positive-mixture module whose output ratio at two control settings is provably monotone in an ordered index such as token distance, retrieval rank, or discretized uncertainty. Use normalized-positive-series identities to replace an unstable quotient derivative with a difference of expectations, and penalize violations of the resulting stochastic-order condition during training.
Insert a fixed polynomial mixer before an MLP or retrieval index for tuples of discrete features. The mixer maps n+2 bounded scalar feature codes to one or several expanded scalar codes, and the paper's theorem guarantees that its image cannot collapse below order |A|^n when all coordinates come from a finite alphabet A. Use multiple independent permutations or coefficient choices to obtain a vector representation while retaining the deterministic algebraic structure.