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We also know that averaging the indices E[i(f,x)] over all functions gives curvature K(x).\n  We explore here the situation when G is geometric of dimension d: that is if each unit sphere S(x) is geometric of dimension d-1 and that X(S(x))=0 for even d and X(S(x))=2 for odd d. The dimension of G is inductively defined as the average of 1+dim(S(x)) over all S(x) assuming the empt"},"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":"1205.0306","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-05-02T02:37:04Z","cross_cats_sorted":["cs.CG","math.GN"],"title_canon_sha256":"47b02a7815c4e46cfb9ba764be44323dc39b22acc5939d87d5f7dd6664b15708","abstract_canon_sha256":"ad0c0f4879470a334d8abcf1d35d92318cf0e721e72b21f90ee91000b21ace4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:34.818891Z","signature_b64":"fWcyaXnPhf4toyVzDapxf1UFFHYS7mEW7RJHyny8Hpslh06HyMACg5reXxykj8omgQhvFtLGOr03NM2VMhYNBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c141b4441a7aaa9ab26efb8c5b420e2c4d700d8d488cffd6e6ffe210c0f5b870","last_reissued_at":"2026-05-18T03:56:34.818370Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:34.818370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An index formula for simple graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.GN"],"primary_cat":"math.DG","authors_text":"Oliver Knill","submitted_at":"2012-05-02T02:37:04Z","abstract_excerpt":"Gauss-Bonnet for simple graphs G assures that the sum of curvatures K(x) over the vertex set V of G is the Euler characteristic X(G). Poincare-Hopf tells that for any injective function f on V the sum of i(f,x) is X(G). We also know that averaging the indices E[i(f,x)] over all functions gives curvature K(x).\n  We explore here the situation when G is geometric of dimension d: that is if each unit sphere S(x) is geometric of dimension d-1 and that X(S(x))=0 for even d and X(S(x))=2 for odd d. The dimension of G is inductively defined as the average of 1+dim(S(x)) over all S(x) assuming the empt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0306","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":""},"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":"1205.0306","created_at":"2026-05-18T03:56:34.818443+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.0306v1","created_at":"2026-05-18T03:56:34.818443+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.0306","created_at":"2026-05-18T03:56:34.818443+00:00"},{"alias_kind":"pith_short_12","alias_value":"YFA3IRA2PKVJ","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"YFA3IRA2PKVJVMTO","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"YFA3IRA2","created_at":"2026-05-18T12:27:27.928770+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/YFA3IRA2PKVJVMTO7OGFWQQOFR","json":"https://pith.science/pith/YFA3IRA2PKVJVMTO7OGFWQQOFR.json","graph_json":"https://pith.science/api/pith-number/YFA3IRA2PKVJVMTO7OGFWQQOFR/graph.json","events_json":"https://pith.science/api/pith-number/YFA3IRA2PKVJVMTO7OGFWQQOFR/events.json","paper":"https://pith.science/paper/YFA3IRA2"},"agent_actions":{"view_html":"https://pith.science/pith/YFA3IRA2PKVJVMTO7OGFWQQOFR","download_json":"https://pith.science/pith/YFA3IRA2PKVJVMTO7OGFWQQOFR.json","view_paper":"https://pith.science/paper/YFA3IRA2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.0306&json=true","fetch_graph":"https://pith.science/api/pith-number/YFA3IRA2PKVJVMTO7OGFWQQOFR/graph.json","fetch_events":"https://pith.science/api/pith-number/YFA3IRA2PKVJVMTO7OGFWQQOFR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YFA3IRA2PKVJVMTO7OGFWQQOFR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YFA3IRA2PKVJVMTO7OGFWQQOFR/action/storage_attestation","attest_author":"https://pith.science/pith/YFA3IRA2PKVJVMTO7OGFWQQOFR/action/author_attestation","sign_citation":"https://pith.science/pith/YFA3IRA2PKVJVMTO7OGFWQQOFR/action/citation_signature","submit_replication":"https://pith.science/pith/YFA3IRA2PKVJVMTO7OGFWQQOFR/action/replication_record"}},"created_at":"2026-05-18T03:56:34.818443+00:00","updated_at":"2026-05-18T03:56:34.818443+00:00"}