arXiv subjects

Levi Segal

Publications and source records attributed to Levi Segal.

3 recordsLinked to original sources

arXiv2026

Obstructions and kernel transport for Hecke lifts of partition q-brackets

By the Bloch-Okounkov theorem, the $q$-bracket sends shifted symmetric functions on partitions to quasimodular forms. Following a question of van Ittersum, we study whether the Hecke action on quasimodular forms lifts through this map, in the sense of exact lifts $A_{n,2k}$ with $\langle A_{n,2k}f\rangle_q = T_{n,2k}\langle f\rangle_q$. Our main theorem is that Zagier's lowering operator $B=\frac{1}{2}(D-\partial^2)$ is injective on the genuine homogeneous subspace in every weight at least $4$, although it is not injective on the formal algebra. For exact lifts compatible with lowering, this transports Hecke actions on $q$-bracket kernels between adjacent weights, with divisibility consequences for characteristic polynomials and $p$-adic constraints on kernel eigenvalues. In fixed weight, we classify all exact lifts by an action on the kernel and a kernel-valued cocycle relative to a section. Two obstruction results show that no exact lift is an algebra homomorphism, and that even strict compatibility with multiplication by $Q_2$ already fails in weight $6$. Finally, we construct exact lifts with scalar kernel action whenever the $q$-bracket image is Hecke-stable, and exact rational computations give $q$-bracket surjectivity through weight $16$ and therefore existence of Hecke lifts through weight $16$.

math.NT↗

arXiv2026

JKO-RAG: Distributional Retrieval as Wasserstein Free-Energy Gradient Flow

RAG pipelines return a \emph{ranked list} of passages. We argue this is a mismatch: the downstream language model conditions on a \emph{set}, and the selection problem is fundamentally geometric. We propose \jko, which frames reranking as minimising a free-energy functional $F(p)=\text{relevance}+\text{entropy}+\text{redundancy}$ under Wasserstein-2 gradient flow via the Jordan--Kinderlehrer--Otto proximal scheme. The ground metric $C_{ij}=(1-\cos\langle z_i,z_j\rangle)^2$ encodes the semantic geometry of the embedding manifold. Our central contribution is a \emph{linear-response theory} explaining \emph{why} the Wasserstein geometry helps: the Wasserstein and KL retrieval maps differ only in their proximal Hessian -- dense and geometry-aware for $W^2$, diagonal and geometry-blind for KL -- and this difference damps the mass transport that query paraphrase induces. The theory yields a falsifiable prediction: the stability advantage is monotonically decreasing in step size $h$. We verify this empirically via free-energy descent, frequency-resolved perturbation response, the predicted $h$-dependence, and a certified-radius analysis. Four extensions are introduced: \textbf{\nmjko} (learned ground metric), \textbf{\bwjko} ($W^2$--KL interpolation), \textbf{\samjko} ($2\times$ speedup), and \textbf{\dualrank} (OT dual potentials as confidence signals). Across five BEIR benchmarks, \jko\ outperforms the cross-encoder on all five; the decisive advantage is robustness -- 22--38\% more stable under paraphrase, $2\times$ fewer leaked distractors.

cs.IR↗

arXiv2026

Contraction Containment in Labeled Trees: Support Counts, Collision Cores, and Star Avoidance

We study whether a labeled tree can be recovered from the numbers of larger labeled trees that contract to it. We first characterize containment using the edge splits induced on the surviving vertices after contraction and derive an exact formula for containing trees with one marked display. We then study several displays in the same containing tree simultaneously. After contracting every edge invisible to all $k$ displays of an $n$-vertex target, the remaining collision core has at most $k(n-1)+1$ vertices. Once the coincidences and relative positions of the surviving labels are recorded, these bounded cores give a finite exact expansion of each collision count. As applications, we prove that almost every sufficiently large labeled tree contains any fixed target, with an exponentially small exceptional proportion. We also obtain an exact avoidance formula and generating function for stars whose center is represented by the largest host label, together with fixed-star asymptotics and a coarse threshold.

math.CO↗