{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:IJT2YYFACNJKAXVMM24VCVMQW7","short_pith_number":"pith:IJT2YYFA","canonical_record":{"source":{"id":"1611.03919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-12T00:26:37Z","cross_cats_sorted":[],"title_canon_sha256":"8f6fe5af428c73e8ef054036d4e2e512ce1e79b6c4a98b8eac1202d6098df707","abstract_canon_sha256":"596cde1d0d74b5fe2514c85a73f00f4791ee05fb9405880fd052d3044e4ffabf"},"schema_version":"1.0"},"canonical_sha256":"4267ac60a01352a05eac66b9515590b7e24845347c87ce1242257fa5ffbe4cbd","source":{"kind":"arxiv","id":"1611.03919","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03919","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03919v1","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03919","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"pith_short_12","alias_value":"IJT2YYFACNJK","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IJT2YYFACNJKAXVM","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IJT2YYFA","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:IJT2YYFACNJKAXVMM24VCVMQW7","target":"record","payload":{"canonical_record":{"source":{"id":"1611.03919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-12T00:26:37Z","cross_cats_sorted":[],"title_canon_sha256":"8f6fe5af428c73e8ef054036d4e2e512ce1e79b6c4a98b8eac1202d6098df707","abstract_canon_sha256":"596cde1d0d74b5fe2514c85a73f00f4791ee05fb9405880fd052d3044e4ffabf"},"schema_version":"1.0"},"canonical_sha256":"4267ac60a01352a05eac66b9515590b7e24845347c87ce1242257fa5ffbe4cbd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:18.079397Z","signature_b64":"eUz6vq2iZ60iumUrgJblqfC1Ptt5hUnfW2F3QI9jsd2PyiCpITCl2tcuPO5GN0TzO2fepNuEy088GAgqehXMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4267ac60a01352a05eac66b9515590b7e24845347c87ce1242257fa5ffbe4cbd","last_reissued_at":"2026-05-18T00:59:18.078776Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:18.078776Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.03919","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:59:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z/qR71OzzRL4k7Dn17lVwSpHk2Mlx9bqvqfWRcDG3fcsOfufrciR/++1zm0DiOKYnX6jbyFrlrZm3cP0oTvNDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T12:20:31.819252Z"},"content_sha256":"711c0fa519b681071905ffb8f84709ce12a84291f82df4d6a61e4720dc933568","schema_version":"1.0","event_id":"sha256:711c0fa519b681071905ffb8f84709ce12a84291f82df4d6a61e4720dc933568"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:IJT2YYFACNJKAXVMM24VCVMQW7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Additive Collatz Trajectories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aalok Thakkar, Mrunmay Jagadale","submitted_at":"2016-11-12T00:26:37Z","abstract_excerpt":"Collatz Conjecture (also known as Ulam's conjecture and 3x+1 problem) concerns the behavior of the iterates of a particular function on natural numbers. A number of generalizations of the conjecture have been subjected to extensive study. This paper explores Additive Collatz Trajectories, a particular case of a generalization of Collatz conjecture and puts forward a sufficient and necessary condition for looping of Additive Collatz Trajectories, along with two minor results. An algorithm to compute the number of equivalence classes when natural numbers are quotiented by the limiting behavior o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03919","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:59:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sOI99JX7VfPs3U430+gPUapKF1ZfoAHKgCsXFz2oKOuG+7SJ8tpCWbkpMipxT0fEGH8g3AoHdL361ww/nilfAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T12:20:31.819929Z"},"content_sha256":"a19bb4d50a45c828d54f6f25e984b8e80401f40309812b9fbbf1942aee5b30af","schema_version":"1.0","event_id":"sha256:a19bb4d50a45c828d54f6f25e984b8e80401f40309812b9fbbf1942aee5b30af"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IJT2YYFACNJKAXVMM24VCVMQW7/bundle.json","state_url":"https://pith.science/pith/IJT2YYFACNJKAXVMM24VCVMQW7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IJT2YYFACNJKAXVMM24VCVMQW7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T12:20:31Z","links":{"resolver":"https://pith.science/pith/IJT2YYFACNJKAXVMM24VCVMQW7","bundle":"https://pith.science/pith/IJT2YYFACNJKAXVMM24VCVMQW7/bundle.json","state":"https://pith.science/pith/IJT2YYFACNJKAXVMM24VCVMQW7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IJT2YYFACNJKAXVMM24VCVMQW7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:IJT2YYFACNJKAXVMM24VCVMQW7","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":"596cde1d0d74b5fe2514c85a73f00f4791ee05fb9405880fd052d3044e4ffabf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-12T00:26:37Z","title_canon_sha256":"8f6fe5af428c73e8ef054036d4e2e512ce1e79b6c4a98b8eac1202d6098df707"},"schema_version":"1.0","source":{"id":"1611.03919","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.03919","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"arxiv_version","alias_value":"1611.03919v1","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.03919","created_at":"2026-05-18T00:59:18Z"},{"alias_kind":"pith_short_12","alias_value":"IJT2YYFACNJK","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IJT2YYFACNJKAXVM","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IJT2YYFA","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:a19bb4d50a45c828d54f6f25e984b8e80401f40309812b9fbbf1942aee5b30af","target":"graph","created_at":"2026-05-18T00:59:18Z","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":"Collatz Conjecture (also known as Ulam's conjecture and 3x+1 problem) concerns the behavior of the iterates of a particular function on natural numbers. A number of generalizations of the conjecture have been subjected to extensive study. This paper explores Additive Collatz Trajectories, a particular case of a generalization of Collatz conjecture and puts forward a sufficient and necessary condition for looping of Additive Collatz Trajectories, along with two minor results. An algorithm to compute the number of equivalence classes when natural numbers are quotiented by the limiting behavior o","authors_text":"Aalok Thakkar, Mrunmay Jagadale","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-12T00:26:37Z","title":"Additive Collatz Trajectories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.03919","kind":"arxiv","version":1},"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":"sha256:711c0fa519b681071905ffb8f84709ce12a84291f82df4d6a61e4720dc933568","target":"record","created_at":"2026-05-18T00:59:18Z","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":"596cde1d0d74b5fe2514c85a73f00f4791ee05fb9405880fd052d3044e4ffabf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-12T00:26:37Z","title_canon_sha256":"8f6fe5af428c73e8ef054036d4e2e512ce1e79b6c4a98b8eac1202d6098df707"},"schema_version":"1.0","source":{"id":"1611.03919","kind":"arxiv","version":1}},"canonical_sha256":"4267ac60a01352a05eac66b9515590b7e24845347c87ce1242257fa5ffbe4cbd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4267ac60a01352a05eac66b9515590b7e24845347c87ce1242257fa5ffbe4cbd","first_computed_at":"2026-05-18T00:59:18.078776Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:59:18.078776Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eUz6vq2iZ60iumUrgJblqfC1Ptt5hUnfW2F3QI9jsd2PyiCpITCl2tcuPO5GN0TzO2fepNuEy088GAgqehXMBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:59:18.079397Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.03919","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:711c0fa519b681071905ffb8f84709ce12a84291f82df4d6a61e4720dc933568","sha256:a19bb4d50a45c828d54f6f25e984b8e80401f40309812b9fbbf1942aee5b30af"],"state_sha256":"99442ba022e4f28df36080dfc0baf3a4cc97e65bcee10ead11ce2d42f1bb1bbb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BmDN/pCG1jcZ9tQbFPytu6akbELlPSzzNo8c49YQ63wJ2Rf2OLxEaQUBjwbygJA2EHIBqZg0KQbRBk4UiriCAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T12:20:31.824019Z","bundle_sha256":"cf604060e67d5802a65a251876ec3e71eca3287cd5223b879ec3cd4675223b10"}}