{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:AGI3VFIDICIBBSQLUJYK3TD637","short_pith_number":"pith:AGI3VFID","schema_version":"1.0","canonical_sha256":"0191ba9503409010ca0ba270adcc7edfcb91c1adebe84b380460c125cafe5a97","source":{"kind":"arxiv","id":"2312.11878","version":4},"attestation_state":"computed","paper":{"title":"Nested homotopy models of finite metric spaces and their spectral homology","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AT","authors_text":"Sergei O. Ivanov","submitted_at":"2023-12-19T06:10:19Z","abstract_excerpt":"For a real $r\\geq 0,$ we consider the notion of $r$-homotopy equivalence in the category quasimetric spaces, which includes metric spaces and directed graphs. We show that for a finite quasimetric space $X$ there is a unique (up to isometry) $r$-homotopy equivalent quasimetric space of the minimal possible cardinality. It is called the $r$-minimal model of $X$. We use this to construct a decomposition of the magnitude-path spectral sequence of a digraph into a direct sum of spectral sequences with certain properties. We also construct an $r$-homotopy invariant ${\\rm SH}^r_{n,I}(X)$ of a quasim"},"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":"2312.11878","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AT","submitted_at":"2023-12-19T06:10:19Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"bdf9627b6bfeedea3d4753b7c762b97f9f8f69beabac7acde7ea82a42aab2c2c","abstract_canon_sha256":"26cd91ab62281589d5dd722fe0e8dd55c215c252cf356ab9285db05c2b6be85e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:21:10.727095Z","signature_b64":"GUq3ifQYStSjnaXTmnYPprB2OXgNkOLeBHhwEv0MDh1X7+Og+6GcDWD4kYo+tfOSQ1xbZs4D8Twzv2K/PBcgDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0191ba9503409010ca0ba270adcc7edfcb91c1adebe84b380460c125cafe5a97","last_reissued_at":"2026-07-05T08:21:10.726639Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:21:10.726639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nested homotopy models of finite metric spaces and their spectral homology","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AT","authors_text":"Sergei O. Ivanov","submitted_at":"2023-12-19T06:10:19Z","abstract_excerpt":"For a real $r\\geq 0,$ we consider the notion of $r$-homotopy equivalence in the category quasimetric spaces, which includes metric spaces and directed graphs. We show that for a finite quasimetric space $X$ there is a unique (up to isometry) $r$-homotopy equivalent quasimetric space of the minimal possible cardinality. It is called the $r$-minimal model of $X$. We use this to construct a decomposition of the magnitude-path spectral sequence of a digraph into a direct sum of spectral sequences with certain properties. We also construct an $r$-homotopy invariant ${\\rm SH}^r_{n,I}(X)$ of a quasim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.11878","kind":"arxiv","version":4},"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/2312.11878/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":"2312.11878","created_at":"2026-07-05T08:21:10.726697+00:00"},{"alias_kind":"arxiv_version","alias_value":"2312.11878v4","created_at":"2026-07-05T08:21:10.726697+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2312.11878","created_at":"2026-07-05T08:21:10.726697+00:00"},{"alias_kind":"pith_short_12","alias_value":"AGI3VFIDICIB","created_at":"2026-07-05T08:21:10.726697+00:00"},{"alias_kind":"pith_short_16","alias_value":"AGI3VFIDICIBBSQL","created_at":"2026-07-05T08:21:10.726697+00:00"},{"alias_kind":"pith_short_8","alias_value":"AGI3VFID","created_at":"2026-07-05T08:21:10.726697+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.15241","citing_title":"Filtered order complexes and magnitude homology of finite graded posets","ref_index":9,"is_internal_anchor":false},{"citing_arxiv_id":"2606.09747","citing_title":"Homotopy theories via the magnitude-path spectral sequence","ref_index":26,"is_internal_anchor":false},{"citing_arxiv_id":"2404.06689","citing_title":"Bigraded path homology and the magnitude-path spectral sequence","ref_index":29,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637","json":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637.json","graph_json":"https://pith.science/api/pith-number/AGI3VFIDICIBBSQLUJYK3TD637/graph.json","events_json":"https://pith.science/api/pith-number/AGI3VFIDICIBBSQLUJYK3TD637/events.json","paper":"https://pith.science/paper/AGI3VFID"},"agent_actions":{"view_html":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637","download_json":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637.json","view_paper":"https://pith.science/paper/AGI3VFID","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2312.11878&json=true","fetch_graph":"https://pith.science/api/pith-number/AGI3VFIDICIBBSQLUJYK3TD637/graph.json","fetch_events":"https://pith.science/api/pith-number/AGI3VFIDICIBBSQLUJYK3TD637/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637/action/storage_attestation","attest_author":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637/action/author_attestation","sign_citation":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637/action/citation_signature","submit_replication":"https://pith.science/pith/AGI3VFIDICIBBSQLUJYK3TD637/action/replication_record"}},"created_at":"2026-07-05T08:21:10.726697+00:00","updated_at":"2026-07-05T08:21:10.726697+00:00"}