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The infamous \\emph{Collatz conjecture} asserts that $\\mathrm{Col}_{\\min}(N)=1$ for all $N \\in \\mathbb{N}+1$. Previously, it was shown by Korec that for any $\\theta > \\frac{\\log 3}{\\log 4} \\approx 0.7924"},"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":"1909.03562","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-09-08T23:15:50Z","cross_cats_sorted":["math.DS","math.NT"],"title_canon_sha256":"947882557cdcebe03e124f89c4867e647c8c6165d7952996aee5be3dad0bc3c0","abstract_canon_sha256":"114db7c451063f7e9187141fb39ccb6d35a416fb1819bc51efe35c0db34cc352"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:57:22.924419Z","signature_b64":"sQvcOdt0QV4g4VmLf7HZyCmgZCf5jmzZBsv/w+wUNRxCLdAlmi9SJmr909lAe2JOCTd6sZWIr479aCU3EZgoAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ae6b2fd02bc2ec4808677df54820f89888ff657894712099c0c12de876a1fbb","last_reissued_at":"2026-07-05T03:57:22.924048Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:57:22.924048Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Almost all orbits of the Collatz map attain almost bounded values","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.NT"],"primary_cat":"math.PR","authors_text":"Terence Tao","submitted_at":"2019-09-08T23:15:50Z","abstract_excerpt":"Define the \\emph{Collatz map} $\\mathrm{Col} : \\mathbb{N}+1 \\to \\mathbb{N}+1$ on the positive integers $\\mathbb{N}+1 = \\{1,2,3,\\dots\\}$ by setting $\\mathrm{Col}(N)$ equal to $3N+1$ when $N$ is odd and $N/2$ when $N$ is even, and let $\\mathrm{Col}_{\\min}(N) := \\inf_{n \\in \\mathbb{N}} \\mathrm{Col}^n(N)$ denote the minimal element of the Collatz orbit $N, \\mathrm{Col}(N), \\mathrm{Col}^2(N), \\dots$. The infamous \\emph{Collatz conjecture} asserts that $\\mathrm{Col}_{\\min}(N)=1$ for all $N \\in \\mathbb{N}+1$. Previously, it was shown by Korec that for any $\\theta > \\frac{\\log 3}{\\log 4} \\approx 0.7924"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.03562","kind":"arxiv","version":5},"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/1909.03562/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":"1909.03562","created_at":"2026-07-05T03:57:22.924104+00:00"},{"alias_kind":"arxiv_version","alias_value":"1909.03562v5","created_at":"2026-07-05T03:57:22.924104+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.03562","created_at":"2026-07-05T03:57:22.924104+00:00"},{"alias_kind":"pith_short_12","alias_value":"HLTLF7ICXQXM","created_at":"2026-07-05T03:57:22.924104+00:00"},{"alias_kind":"pith_short_16","alias_value":"HLTLF7ICXQXMJAEG","created_at":"2026-07-05T03:57:22.924104+00:00"},{"alias_kind":"pith_short_8","alias_value":"HLTLF7IC","created_at":"2026-07-05T03:57:22.924104+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.13886","citing_title":"Parity vectors and paradoxical sequences in the accelerated Collatz map","ref_index":2,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG","json":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG.json","graph_json":"https://pith.science/api/pith-number/HLTLF7ICXQXMJAEGO7PVJAQPRG/graph.json","events_json":"https://pith.science/api/pith-number/HLTLF7ICXQXMJAEGO7PVJAQPRG/events.json","paper":"https://pith.science/paper/HLTLF7IC"},"agent_actions":{"view_html":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG","download_json":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG.json","view_paper":"https://pith.science/paper/HLTLF7IC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1909.03562&json=true","fetch_graph":"https://pith.science/api/pith-number/HLTLF7ICXQXMJAEGO7PVJAQPRG/graph.json","fetch_events":"https://pith.science/api/pith-number/HLTLF7ICXQXMJAEGO7PVJAQPRG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG/action/storage_attestation","attest_author":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG/action/author_attestation","sign_citation":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG/action/citation_signature","submit_replication":"https://pith.science/pith/HLTLF7ICXQXMJAEGO7PVJAQPRG/action/replication_record"}},"created_at":"2026-07-05T03:57:22.924104+00:00","updated_at":"2026-07-05T03:57:22.924104+00:00"}