{"paper":{"title":"Box Progressions, Abelian Power-Free Morphisms and A Sieve Technique for the Template Method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haydar G\\\"oral, Nihan Tan{\\i}sal{\\i}, Sad{\\i}k Eyido\\u{g}an","submitted_at":"2026-05-19T21:21:46Z","abstract_excerpt":"Given balls and boxes both enumerated by the positive integers, we consider a sequential allocation of the balls into the boxes. We fix $\\ell \\ge 2$. Proceeding in increasing order of box labels, assign to each box the next $r$ smallest balls for some $ 1\\leq r\\leq\\ell$. Given an integer $k\\ge 3$, is there a natural number $N$ such that in any placement of $N$ balls into boxes, there exist $k$ balls whose labels and box labels each form a $k$-term arithmetic progression? We address this question by identifying abelian power-free fixed points of morphisms over a binary alphabet. We present suff"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20504","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.20504/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}