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The function energizes the geometry similarly as divisors do in the continuum, where the Riemann-Roch quantity chi(G)+deg(D) plays the role of the energy. Define the n times n matrices L=L^--(x,y)=E[W^-(x) cap W^-(y)] and L^++(x,y) = E[W^+(x) cap W^+(y)]. With the notation S(x,y)=1_n omega(x) =delta(x,y) (-1)dim(x) and str(A)=tr(SA) define g=S L^++ S. The results are: "},"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":"1908.06563","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-19T02:40:27Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"d720e2378a0253edc5b3697449a16e2f16374cbba503a84c5beb6ea7a64d384e","abstract_canon_sha256":"02f7dba00a68cb61241f565bd5bc1d40bec65787702cfc1d0562620a189f2f38"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:58:17.204818Z","signature_b64":"VwpcdEYX3fVoCFJOGVNqSI2rUzaSwLaW13Wg9FxT7uBWVOMB/TKzUSmrvqOFxPqk9Ds2hEBPnU4cJ6uHj51XDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a51dbfc73d0f772d0f6ef7fe47f4bd8e72e7b5f9fd381a3f284c76019dbb53a","last_reissued_at":"2026-07-04T23:58:17.204465Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:58:17.204465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Energized simplicial complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Oliver Knill","submitted_at":"2019-08-19T02:40:27Z","abstract_excerpt":"For a simplicial complex with n sets, let W^-(x) be the set of sets in G contained in x and W^+(x) the set of sets in G containing x. An integer-valued function h on G defines for every A subset G an energy E[A]=sum_x in A h(x). The function energizes the geometry similarly as divisors do in the continuum, where the Riemann-Roch quantity chi(G)+deg(D) plays the role of the energy. Define the n times n matrices L=L^--(x,y)=E[W^-(x) cap W^-(y)] and L^++(x,y) = E[W^+(x) cap W^+(y)]. With the notation S(x,y)=1_n omega(x) =delta(x,y) (-1)dim(x) and str(A)=tr(SA) define g=S L^++ S. The results are: "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06563","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/1908.06563/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":"1908.06563","created_at":"2026-07-04T23:58:17.204528+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.06563v1","created_at":"2026-07-04T23:58:17.204528+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.06563","created_at":"2026-07-04T23:58:17.204528+00:00"},{"alias_kind":"pith_short_12","alias_value":"FJI5X7DT2D3X","created_at":"2026-07-04T23:58:17.204528+00:00"},{"alias_kind":"pith_short_16","alias_value":"FJI5X7DT2D3XFUHW","created_at":"2026-07-04T23:58:17.204528+00:00"},{"alias_kind":"pith_short_8","alias_value":"FJI5X7DT","created_at":"2026-07-04T23:58:17.204528+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2508.14372","citing_title":"Dehn Sommerville Manifolds","ref_index":30,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D","json":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D.json","graph_json":"https://pith.science/api/pith-number/FJI5X7DT2D3XFUHW5576I72L3D/graph.json","events_json":"https://pith.science/api/pith-number/FJI5X7DT2D3XFUHW5576I72L3D/events.json","paper":"https://pith.science/paper/FJI5X7DT"},"agent_actions":{"view_html":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D","download_json":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D.json","view_paper":"https://pith.science/paper/FJI5X7DT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.06563&json=true","fetch_graph":"https://pith.science/api/pith-number/FJI5X7DT2D3XFUHW5576I72L3D/graph.json","fetch_events":"https://pith.science/api/pith-number/FJI5X7DT2D3XFUHW5576I72L3D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D/action/storage_attestation","attest_author":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D/action/author_attestation","sign_citation":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D/action/citation_signature","submit_replication":"https://pith.science/pith/FJI5X7DT2D3XFUHW5576I72L3D/action/replication_record"}},"created_at":"2026-07-04T23:58:17.204528+00:00","updated_at":"2026-07-04T23:58:17.204528+00:00"}