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A metric space is called $\\Delta$-generic, if the set of Gromov products at each $P_i$ has a unique smallest element $\\Delta_{ijk}$.\n  For a $\\Delta$-generic metric space, the map $P_i\\to (P_jP_k)$, where $(P_jP_k)$ is the edge joining $P_j$ to $P_k$ is a well defined map called the \"Gromov product structure\" [Bilge, Celik and Kocak, \"A"},"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":"1804.03051","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-09T15:11:12Z","cross_cats_sorted":[],"title_canon_sha256":"ec434a314226b41da17e9c314ace65e5f485d0d4f79fd76bc985916628e8a336","abstract_canon_sha256":"5001ea8672620962ae23d244833bd83f8b5ef47d9e39e9b596a6debadc00a9f6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:57.529391Z","signature_b64":"Svv0/lOB8RRcmWjneq/ywZlCQ3PgGnDpdLBT3ylepWOnYBZqeWPWY0KdycGgLd3cp6pY4ki4Ov3TF0Hjn4cFBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2cb03b11e4ec1b6c5fd29e7bbf6280950dce3210a2a8aa59672f3b9a3e48be9","last_reissued_at":"2026-05-18T00:18:57.528711Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:57.528711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gromov Product Decomposition of 7-point Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ayse Humeyra Bilge, Metehan Incegul","submitted_at":"2018-04-09T15:11:12Z","abstract_excerpt":"Let $X$ be a finite metric space with elements $P_i$, $i=1,\\dots,n$ and with distance functions $d_{ij}$. The Gromov product of the triangle with vertices $P_i$, $P_j$ and $P_k$ at the vertex $P_i$ is defined by $\\Delta_{ijk}=\\frac{1}{2}(d_{ij}+d_{ik}-d_{jk})$. A metric space is called $\\Delta$-generic, if the set of Gromov products at each $P_i$ has a unique smallest element $\\Delta_{ijk}$.\n  For a $\\Delta$-generic metric space, the map $P_i\\to (P_jP_k)$, where $(P_jP_k)$ is the edge joining $P_j$ to $P_k$ is a well defined map called the \"Gromov product structure\" [Bilge, Celik and Kocak, \"A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03051","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":"1804.03051","created_at":"2026-05-18T00:18:57.528815+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.03051v1","created_at":"2026-05-18T00:18:57.528815+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.03051","created_at":"2026-05-18T00:18:57.528815+00:00"},{"alias_kind":"pith_short_12","alias_value":"6LFQHMI6J3A3","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"6LFQHMI6J3A3NRP5","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"6LFQHMI6","created_at":"2026-05-18T12:32:08.215937+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/6LFQHMI6J3A3NRP5FHT3X5RIBF","json":"https://pith.science/pith/6LFQHMI6J3A3NRP5FHT3X5RIBF.json","graph_json":"https://pith.science/api/pith-number/6LFQHMI6J3A3NRP5FHT3X5RIBF/graph.json","events_json":"https://pith.science/api/pith-number/6LFQHMI6J3A3NRP5FHT3X5RIBF/events.json","paper":"https://pith.science/paper/6LFQHMI6"},"agent_actions":{"view_html":"https://pith.science/pith/6LFQHMI6J3A3NRP5FHT3X5RIBF","download_json":"https://pith.science/pith/6LFQHMI6J3A3NRP5FHT3X5RIBF.json","view_paper":"https://pith.science/paper/6LFQHMI6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.03051&json=true","fetch_graph":"https://pith.science/api/pith-number/6LFQHMI6J3A3NRP5FHT3X5RIBF/graph.json","fetch_events":"https://pith.science/api/pith-number/6LFQHMI6J3A3NRP5FHT3X5RIBF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6LFQHMI6J3A3NRP5FHT3X5RIBF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6LFQHMI6J3A3NRP5FHT3X5RIBF/action/storage_attestation","attest_author":"https://pith.science/pith/6LFQHMI6J3A3NRP5FHT3X5RIBF/action/author_attestation","sign_citation":"https://pith.science/pith/6LFQHMI6J3A3NRP5FHT3X5RIBF/action/citation_signature","submit_replication":"https://pith.science/pith/6LFQHMI6J3A3NRP5FHT3X5RIBF/action/replication_record"}},"created_at":"2026-05-18T00:18:57.528815+00:00","updated_at":"2026-05-18T00:18:57.528815+00:00"}