{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:MQHQG2RO6CLDMJE3X7X7I5KFW3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5028549c8423813084ed4db288830343ac3b076f978b48bcc09fbabf7e3b810e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-14T21:37:14Z","title_canon_sha256":"5781c829b69bdf5bd30035294741265afe6b4ebae47c904eb79540b45a6ff089"},"schema_version":"1.0","source":{"id":"1907.07088","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.07088","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"arxiv_version","alias_value":"1907.07088v2","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.07088","created_at":"2026-05-17T23:40:22Z"},{"alias_kind":"pith_short_12","alias_value":"MQHQG2RO6CLD","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"MQHQG2RO6CLDMJE3","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"MQHQG2RO","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:93674ea0f019b93180a9cf426c8d88527f85b2ae335d02eaa01a66f4929f7f4a","target":"graph","created_at":"2026-05-17T23:40:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The Collatz conjecture asserts that repeatedly iterating $f(x) = (3x + 1)/2^{a(x)}$, where $a(x)$ is the highest exponent for which $2^{a(x)}$ exactly divides $3x+1$, always lead to $1$ for any odd positive integer $x$. Here, we present an arborescence graph constructed from iterations of $g(x) = (2^{e(x)}x - 1)/3$, which is the inverse of $f(x)$ and where $x \\not \\equiv [0]_3$ and $e(x)$ is any positive integer satisfying $2^{e(x)}x - 1 \\equiv [0]_3$, with $[0]_3$ denoting $0\\pmod{3}$. The integer patterns inferred from the resulting arborescence provide new insights into proving the validity","authors_text":"Zenon B. Batang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-14T21:37:14Z","title":"Integer patterns in Collatz sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.07088","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c3602438b71e541d58d30a13d54b8a3d2dbc4a1f9355fabad9fbf47c25306b48","target":"record","created_at":"2026-05-17T23:40:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5028549c8423813084ed4db288830343ac3b076f978b48bcc09fbabf7e3b810e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-14T21:37:14Z","title_canon_sha256":"5781c829b69bdf5bd30035294741265afe6b4ebae47c904eb79540b45a6ff089"},"schema_version":"1.0","source":{"id":"1907.07088","kind":"arxiv","version":2}},"canonical_sha256":"640f036a2ef09636249bbfeff47545b6df46600a4b0af4601788882995481170","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"640f036a2ef09636249bbfeff47545b6df46600a4b0af4601788882995481170","first_computed_at":"2026-05-17T23:40:22.924932Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:22.924932Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qS/+pB5UogHx5TZbt+WZbpECgGDDENfskLcMmUdwAQs1MwI4yW5whPWQERAEyUON0N3vK6Qn8wcc7+LKnhh8Cg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:22.925567Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.07088","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3602438b71e541d58d30a13d54b8a3d2dbc4a1f9355fabad9fbf47c25306b48","sha256:93674ea0f019b93180a9cf426c8d88527f85b2ae335d02eaa01a66f4929f7f4a"],"state_sha256":"3776d2f7c406716b0a56459fe87b4eaa28c4da4f619cb5ec152f9ae9b54fed51"}