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We prove that, in every finite colouring of $\\mathbb{R}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \\[\n  (c_2(A_{\\mathbf v}),\\ldots,c_d(A_{\\mathbf v})). \\] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\\mathbf v})=\\det(A_{\\mathbf v})$. We also prove a discrete analogue: if $E\\subseteq\\mathbb{Z}^d$ has positive upper Banach density, then, for some $"},"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":"2606.12947","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DS","submitted_at":"2026-06-11T06:20:02Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"fe2ca4f7d7ab491b44833f2ed99f1e02926bf5dae7bfa99b09d5b56dcf915b8f","abstract_canon_sha256":"da12522d774ffe8c31f2a8a73578d896cbe4cf1d5ec04324f205a5d4cd755d7c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:09:35.224378Z","signature_b64":"n77iOV4eMUMjRi4GUJ8nOGHxyxgR32oMxjVYRVaLHQlrHw6zJZledCYUSwktzj+QFWtAC+IPcb1gi9giZIhWCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"adce9ccaecf76021375151d7cd2b55d1799369dd0d54f77c78047ce7aca1e1ca","last_reissued_at":"2026-06-12T01:09:35.223987Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:09:35.223987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Trace spectra of simplices in large sets","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.DS","authors_text":"Alexander Fish, Michael Bj\\\"orklund, Shrey Sanadhya","submitted_at":"2026-06-11T06:20:02Z","abstract_excerpt":"Given an ordered tuple $\\mathbf v=(v_0,\\ldots,v_d)$ of vectors in $\\mathbb{R}^d$, let $A_{\\mathbf v}=[\\,v_1-v_0\\ \\cdots\\ v_d-v_0\\,]$ be its edge matrix. We prove that, in every finite colouring of $\\mathbb{R}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \\[\n  (c_2(A_{\\mathbf v}),\\ldots,c_d(A_{\\mathbf v})). \\] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\\mathbf v})=\\det(A_{\\mathbf v})$. We also prove a discrete analogue: if $E\\subseteq\\mathbb{Z}^d$ has positive upper Banach density, then, for some $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12947","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/2606.12947/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":"2606.12947","created_at":"2026-06-12T01:09:35.224047+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.12947v1","created_at":"2026-06-12T01:09:35.224047+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.12947","created_at":"2026-06-12T01:09:35.224047+00:00"},{"alias_kind":"pith_short_12","alias_value":"VXHJZSXM65QC","created_at":"2026-06-12T01:09:35.224047+00:00"},{"alias_kind":"pith_short_16","alias_value":"VXHJZSXM65QCCN2R","created_at":"2026-06-12T01:09:35.224047+00:00"},{"alias_kind":"pith_short_8","alias_value":"VXHJZSXM","created_at":"2026-06-12T01:09:35.224047+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F","json":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F.json","graph_json":"https://pith.science/api/pith-number/VXHJZSXM65QCCN2RKHL42K2V2F/graph.json","events_json":"https://pith.science/api/pith-number/VXHJZSXM65QCCN2RKHL42K2V2F/events.json","paper":"https://pith.science/paper/VXHJZSXM"},"agent_actions":{"view_html":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F","download_json":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F.json","view_paper":"https://pith.science/paper/VXHJZSXM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.12947&json=true","fetch_graph":"https://pith.science/api/pith-number/VXHJZSXM65QCCN2RKHL42K2V2F/graph.json","fetch_events":"https://pith.science/api/pith-number/VXHJZSXM65QCCN2RKHL42K2V2F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F/action/storage_attestation","attest_author":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F/action/author_attestation","sign_citation":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F/action/citation_signature","submit_replication":"https://pith.science/pith/VXHJZSXM65QCCN2RKHL42K2V2F/action/replication_record"}},"created_at":"2026-06-12T01:09:35.224047+00:00","updated_at":"2026-06-12T01:09:35.224047+00:00"}