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We find a different layered permutation $w\\in S_n$ whose Grothendieck polynomial has support of size asymptotically at least $n!/e^{\\sqrt{2n} \\cdot \\ln(n)}$ and obtain more precise asymptotics for the growth rate of $\\beta^{\\mathfrak G}(n):=\\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak G_w)|$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2512.04053","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-12-03T18:36:13Z","cross_cats_sorted":[],"title_canon_sha256":"a02a3561728684c48bc2059d72c1ee315492da01ae197db438993fb354d96f92","abstract_canon_sha256":"0cff905dc50fc4182d70887a633151ba6e88775976e304875a037ef0d2542adb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:11:18.002912Z","signature_b64":"nmXuWbTjPUCpS0gC4e01l46ICji+cSXSoJhgz+ufWCbEKqy8dk9IoSbEojNRZhMK6/0lvtO+jrobtW35o6cMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16ab26ee09d12d04395aace93a829c143aea794d19ec5c57450fa163b7026b98","last_reissued_at":"2026-06-19T16:11:18.002443Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:11:18.002443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotically maximal Schubitopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jack Chen-An Chou, Linus Setiabrata","submitted_at":"2025-12-03T18:36:13Z","abstract_excerpt":"We find a layered permutation $w\\in S_n$ whose Schubert polynomial $\\mathfrak S_w(x_1, \\dots, x_n)$ has support of size asymptotically at least $n!/4^n$. This gives precise asymptotics for the growth rate of $\\beta(n):= \\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak S_w)|$. We find a different layered permutation $w\\in S_n$ whose Grothendieck polynomial has support of size asymptotically at least $n!/e^{\\sqrt{2n} \\cdot \\ln(n)}$ and obtain more precise asymptotics for the growth rate of $\\beta^{\\mathfrak G}(n):=\\max_{w\\in S_n}|\\mathrm{supp}(\\mathfrak G_w)|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.04053","kind":"arxiv","version":2},"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/2512.04053/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2512.04053","created_at":"2026-06-19T16:11:18.002501+00:00"},{"alias_kind":"arxiv_version","alias_value":"2512.04053v2","created_at":"2026-06-19T16:11:18.002501+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.04053","created_at":"2026-06-19T16:11:18.002501+00:00"},{"alias_kind":"pith_short_12","alias_value":"C2VSN3QJ2EWQ","created_at":"2026-06-19T16:11:18.002501+00:00"},{"alias_kind":"pith_short_16","alias_value":"C2VSN3QJ2EWQIOK2","created_at":"2026-06-19T16:11:18.002501+00:00"},{"alias_kind":"pith_short_8","alias_value":"C2VSN3QJ","created_at":"2026-06-19T16:11:18.002501+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.10276","citing_title":"Principal specializations of Grothendieck polynomials","ref_index":4,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ","json":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ.json","graph_json":"https://pith.science/api/pith-number/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/graph.json","events_json":"https://pith.science/api/pith-number/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/events.json","paper":"https://pith.science/paper/C2VSN3QJ"},"agent_actions":{"view_html":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ","download_json":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ.json","view_paper":"https://pith.science/paper/C2VSN3QJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2512.04053&json=true","fetch_graph":"https://pith.science/api/pith-number/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/graph.json","fetch_events":"https://pith.science/api/pith-number/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/action/storage_attestation","attest_author":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/action/author_attestation","sign_citation":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/action/citation_signature","submit_replication":"https://pith.science/pith/C2VSN3QJ2EWQIOK2VTUTVAU4CQ/action/replication_record"}},"created_at":"2026-06-19T16:11:18.002501+00:00","updated_at":"2026-06-19T16:11:18.002501+00:00"}