{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2005:K36ONKAIKK7Z6XOT4N3PWDT5YX","short_pith_number":"pith:K36ONKAI","schema_version":"1.0","canonical_sha256":"56fce6a80852bf9f5dd3e376fb0e7dc5e09ab07dfbbe19a4742057584c044716","source":{"kind":"arxiv","id":"math/0503274","version":1},"attestation_state":"computed","paper":{"title":"Flows and joins of metric spaces","license":"","headline":"","cross_cats":["math.GR","math.GT"],"primary_cat":"math.MG","authors_text":"Igor Mineyev","submitted_at":"2005-03-14T21:21:00Z","abstract_excerpt":"We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X.\n  Classical concepts known for H^n and negatively curved manifolds are defined in a precise way for any hyperbolic complex X, for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification X-bar= X union bdry X. They are cont"},"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":"math/0503274","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"2005-03-14T21:21:00Z","cross_cats_sorted":["math.GR","math.GT"],"title_canon_sha256":"2d38bc52f10adec66ce4e7bcb24b18f7761c3d8fc3df525fc910d6adc51dbc7c","abstract_canon_sha256":"512a04775fa3465705ba5a53f9798c40e61c8656e45012f95a7b28fb77818c10"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:37:59.792776Z","signature_b64":"XQYkwjgIPm/UDI0mmJ0BSG88As3PbnTqQqqZmqCU3sfoB+N3wphNUWUEXC8mtZRSDjvUegI1bFln4HD+7UJDCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56fce6a80852bf9f5dd3e376fb0e7dc5e09ab07dfbbe19a4742057584c044716","last_reissued_at":"2026-05-18T02:37:59.792401Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:37:59.792401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Flows and joins of metric spaces","license":"","headline":"","cross_cats":["math.GR","math.GT"],"primary_cat":"math.MG","authors_text":"Igor Mineyev","submitted_at":"2005-03-14T21:21:00Z","abstract_excerpt":"We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X.\n  Classical concepts known for H^n and negatively curved manifolds are defined in a precise way for any hyperbolic complex X, for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification X-bar= X union bdry X. They are cont"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0503274","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":"math/0503274","created_at":"2026-05-18T02:37:59.792459+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0503274v1","created_at":"2026-05-18T02:37:59.792459+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0503274","created_at":"2026-05-18T02:37:59.792459+00:00"},{"alias_kind":"pith_short_12","alias_value":"K36ONKAIKK7Z","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"K36ONKAIKK7Z6XOT","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"K36ONKAI","created_at":"2026-05-18T12:25:53.335082+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/K36ONKAIKK7Z6XOT4N3PWDT5YX","json":"https://pith.science/pith/K36ONKAIKK7Z6XOT4N3PWDT5YX.json","graph_json":"https://pith.science/api/pith-number/K36ONKAIKK7Z6XOT4N3PWDT5YX/graph.json","events_json":"https://pith.science/api/pith-number/K36ONKAIKK7Z6XOT4N3PWDT5YX/events.json","paper":"https://pith.science/paper/K36ONKAI"},"agent_actions":{"view_html":"https://pith.science/pith/K36ONKAIKK7Z6XOT4N3PWDT5YX","download_json":"https://pith.science/pith/K36ONKAIKK7Z6XOT4N3PWDT5YX.json","view_paper":"https://pith.science/paper/K36ONKAI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0503274&json=true","fetch_graph":"https://pith.science/api/pith-number/K36ONKAIKK7Z6XOT4N3PWDT5YX/graph.json","fetch_events":"https://pith.science/api/pith-number/K36ONKAIKK7Z6XOT4N3PWDT5YX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K36ONKAIKK7Z6XOT4N3PWDT5YX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K36ONKAIKK7Z6XOT4N3PWDT5YX/action/storage_attestation","attest_author":"https://pith.science/pith/K36ONKAIKK7Z6XOT4N3PWDT5YX/action/author_attestation","sign_citation":"https://pith.science/pith/K36ONKAIKK7Z6XOT4N3PWDT5YX/action/citation_signature","submit_replication":"https://pith.science/pith/K36ONKAIKK7Z6XOT4N3PWDT5YX/action/replication_record"}},"created_at":"2026-05-18T02:37:59.792459+00:00","updated_at":"2026-05-18T02:37:59.792459+00:00"}