arXiv:2609. 29881v1 Announce Type: cross Abstract: Choosing Shellsort gaps is a well-known open problem.
arXiv:2606. 13799v1 Announce Type: cross Abstract: Finding the shortest program that generates a sequence is uncomputable, and for six decades that fact has been mistaken for a wall around finding any generating program.
arXiv:2606. 03419v1 Announce Type: cross Abstract: The 2026 disproof of Erd\H{o}s's unit-distance conjecture and Sawin's subsequent explicit quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$.
arXiv:2609.13692v1 Announce Type: cross Abstract: LLM serving reuses KV cache by exact prefix match, so when a prompt is assembled from a set of reusable pieces -- retrieved passages, tool definition...
arXiv:2606. 26399v1 Announce Type: new Abstract: We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints.
We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits.
The paper presents a table‑free index for tapered memoization grids, enabling compact out‑of‑core evaluation of functions that depend on sorted arguments. By showing that the grid’s key set corresponds to multiset combinations, the authors derive a closed‑form O(d) ranking and unranking scheme that removes the need for large preprocessing tables and allows order‑free parallel construction. The resulting values‑only flat array uses significantly less memory than hash‑map memoization, offers faster query times once cache limits are exceeded, and remains operable with memory‑mapped storage beyond RAM.
The paper reports on a large‑scale verified search experiment using a 30B language model on a laptop, evaluating three operator packages—schematic notebooks, named obstacles, and behavioural repulsion—in a factorial design across nine construction problems. Results show that the full composition of operators closes the seed‑to‑record gap more effectively than any single component, increases construction‑hash diversity, and that memory plus repulsion consistently avoids collapse. A frontier proposer achieves similar gains in far fewer samples, but the search ultimately stalls near a plateau where the reference family is adopted and optimized only when provided as code.
arXiv:2606. 25777v1 Announce Type: cross Abstract: We initiate a resource-aware theory of \textit{language generation in the limit} under the minimal constraint of space efficiency.
arXiv:2608. 06825v1 Announce Type: new Abstract: Learning from correct demonstrations is harder than supervised learning when many answers are correct: after predicting, the learner sees one valid answer but not whether its own answer was valid, nor any reward.
arXiv:2609. 04046v1 Announce Type: cross Abstract: What can a single layer of self-attention compute?
arXiv:2606. 29082v1 Announce Type: cross Abstract: Would experience designing faster GPU kernels also help close in on a long-standing open mathematical conjecture?
arXiv:2607. 17481v1 Announce Type: new Abstract: Motif discovery, the search for recurring patterns within a time series, is a core primitive of exploratory data analysis.