{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:Q5ARZXTQCWRYSGIOAICFIHPVHN","short_pith_number":"pith:Q5ARZXTQ","schema_version":"1.0","canonical_sha256":"87411cde7015a389190e0204541df53b75270901692ede074f1d040082a73486","source":{"kind":"arxiv","id":"2504.14629","version":1},"attestation_state":"computed","paper":{"title":"New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Ivan N. Mikhailov","submitted_at":"2025-04-20T14:11:57Z","abstract_excerpt":"In the paper we prove that, for arbitrary unbounded subset $A\\subset R$ and an arbitrary bounded metric space~$X$, a curve $A\\times_{\\ell^1} (tX)$, $t\\in[0,\\,\\infty)$ is a geodesic line in the Gromov--Hausdorff class. We also show that, for abitrary $\\lambda > 1$, $n\\in\\mathbb{N}$, the following inequality holds: $d_{GH}\\bigl(\\mathbb{Z}^n,\\,\\lambda\\mathbb{Z}^n\\bigr)\\ge\\frac{1}{2}$. We conclude that a curve $t\\mathbb{Z}^n$, $t\\in(0,\\,\\infty)$ is not continuous with respect to the Gromov--Hausdorff distance, and, therefore, is not a gedesic line. Moreover, it follows that multiplication of all m"},"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":"2504.14629","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-04-20T14:11:57Z","cross_cats_sorted":[],"title_canon_sha256":"6613ec8797c7fb71f684ebec3e40a9dae86551576ca6cb86cee0ba706f2af1a2","abstract_canon_sha256":"4d4bafae356d1e51db7791946126c2ecb7bfbf919389927c1583f0b662e8b455"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:51:47.858331Z","signature_b64":"wkxVKE0ZAtEFIqjoKLe740i3+r5qBXHlUv1LEtx82iUczMLpi1QJlqMd5uwcNdXdOkG3M3EEnbd2YShGetOfAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87411cde7015a389190e0204541df53b75270901692ede074f1d040082a73486","last_reissued_at":"2026-07-05T10:51:47.857793Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:51:47.857793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Ivan N. Mikhailov","submitted_at":"2025-04-20T14:11:57Z","abstract_excerpt":"In the paper we prove that, for arbitrary unbounded subset $A\\subset R$ and an arbitrary bounded metric space~$X$, a curve $A\\times_{\\ell^1} (tX)$, $t\\in[0,\\,\\infty)$ is a geodesic line in the Gromov--Hausdorff class. We also show that, for abitrary $\\lambda > 1$, $n\\in\\mathbb{N}$, the following inequality holds: $d_{GH}\\bigl(\\mathbb{Z}^n,\\,\\lambda\\mathbb{Z}^n\\bigr)\\ge\\frac{1}{2}$. We conclude that a curve $t\\mathbb{Z}^n$, $t\\in(0,\\,\\infty)$ is not continuous with respect to the Gromov--Hausdorff distance, and, therefore, is not a gedesic line. Moreover, it follows that multiplication of all m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.14629","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/2504.14629/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":"2504.14629","created_at":"2026-07-05T10:51:47.857860+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.14629v1","created_at":"2026-07-05T10:51:47.857860+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.14629","created_at":"2026-07-05T10:51:47.857860+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q5ARZXTQCWRY","created_at":"2026-07-05T10:51:47.857860+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q5ARZXTQCWRYSGIO","created_at":"2026-07-05T10:51:47.857860+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q5ARZXTQ","created_at":"2026-07-05T10:51:47.857860+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.19680","citing_title":"Contractibility of the space of $\\varepsilon$-nets in $\\mathbb{R}$","ref_index":6,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN","json":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN.json","graph_json":"https://pith.science/api/pith-number/Q5ARZXTQCWRYSGIOAICFIHPVHN/graph.json","events_json":"https://pith.science/api/pith-number/Q5ARZXTQCWRYSGIOAICFIHPVHN/events.json","paper":"https://pith.science/paper/Q5ARZXTQ"},"agent_actions":{"view_html":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN","download_json":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN.json","view_paper":"https://pith.science/paper/Q5ARZXTQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.14629&json=true","fetch_graph":"https://pith.science/api/pith-number/Q5ARZXTQCWRYSGIOAICFIHPVHN/graph.json","fetch_events":"https://pith.science/api/pith-number/Q5ARZXTQCWRYSGIOAICFIHPVHN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN/action/storage_attestation","attest_author":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN/action/author_attestation","sign_citation":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN/action/citation_signature","submit_replication":"https://pith.science/pith/Q5ARZXTQCWRYSGIOAICFIHPVHN/action/replication_record"}},"created_at":"2026-07-05T10:51:47.857860+00:00","updated_at":"2026-07-05T10:51:47.857860+00:00"}