{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:QPDTWC7A4DHFMTBF3G2OGHPGIM","short_pith_number":"pith:QPDTWC7A","canonical_record":{"source":{"id":"2605.24909","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-24T07:30:03Z","cross_cats_sorted":[],"title_canon_sha256":"f004434d35b0899dc8a81511215ba05c277ba5795977d79362c4286fa29488c5","abstract_canon_sha256":"a9163815129559208d2e7e082f10b423f59f967c94b4d8156337111b624283c5"},"schema_version":"1.0"},"canonical_sha256":"83c73b0be0e0ce564c25d9b4e31de6433488c58e90434488e7b240bd45276d91","source":{"kind":"arxiv","id":"2605.24909","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.24909","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"arxiv_version","alias_value":"2605.24909v1","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.24909","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"pith_short_12","alias_value":"QPDTWC7A4DHF","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"pith_short_16","alias_value":"QPDTWC7A4DHFMTBF","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"pith_short_8","alias_value":"QPDTWC7A","created_at":"2026-05-26T01:04:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:QPDTWC7A4DHFMTBF3G2OGHPGIM","target":"record","payload":{"canonical_record":{"source":{"id":"2605.24909","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-24T07:30:03Z","cross_cats_sorted":[],"title_canon_sha256":"f004434d35b0899dc8a81511215ba05c277ba5795977d79362c4286fa29488c5","abstract_canon_sha256":"a9163815129559208d2e7e082f10b423f59f967c94b4d8156337111b624283c5"},"schema_version":"1.0"},"canonical_sha256":"83c73b0be0e0ce564c25d9b4e31de6433488c58e90434488e7b240bd45276d91","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:04:05.184177Z","signature_b64":"1GLT3nJvHe1Vww3tbOY3BnmcwB2kYXnEo38+o4JUMEXeENgttQGfqMR6pkMWFUqneIZBZBI6G7ePBMMX9ZqfAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"83c73b0be0e0ce564c25d9b4e31de6433488c58e90434488e7b240bd45276d91","last_reissued_at":"2026-05-26T01:04:05.183393Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:04:05.183393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.24909","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":"2026-05-26T01:04:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B5JTv+EXMqkHuMh8RI3ymEwbFhJ0Zzj5K9ZZeom5FkGKd7NRL9rf+f4kFn4sX3+Rlps7AKLy2CnN985HTd4CBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T12:42:27.454195Z"},"content_sha256":"b10ee1042dd36ed242923705c4ad349f073e83949f7e894da547bcd617dd09ae","schema_version":"1.0","event_id":"sha256:b10ee1042dd36ed242923705c4ad349f073e83949f7e894da547bcd617dd09ae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:QPDTWC7A4DHFMTBF3G2OGHPGIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Valuation Separation for Coprime Lucas Products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dongyeon Kym","submitted_at":"2026-05-24T07:30:03Z","abstract_excerpt":"Let $U_n=U_n(P,Q)$ be a nondegenerate Lucas sequence with $Q=\\pm 1$ and discriminant $\\Delta=P^2+4Q>0$. We study Diophantine equations \\[ A y^k=\\prod_{i=1}^r U_{n_i}(P,Q), \\qquad k\\geq 2, \\] where the indices $n_1,\\ldots,n_r$ are pairwise coprime. The strong divisibility property implies that the factors $U_{n_i}$ are pairwise coprime, and hence a global $k$-th power condition separates into local valuation conditions on the individual factors. For $k=2$, this gives a termwise square-class restriction: each $U_{n_i}$ has signed squarefree part supported on the primes dividing $A$. In particula"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24909","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/2605.24909/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-26T01:04:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rq5UgVcYzTIiGw/b7Z2LNNVwbUOEvTORxzaWeAf+BF/K7R+1effQ+wYGiskUvnIyHHyIWfg5BPRjk22ptV+QDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T12:42:27.454703Z"},"content_sha256":"8241732fae22a7a36db7fd27a0fd1d8d52ec550240ca78a584a9fe5858c0d39e","schema_version":"1.0","event_id":"sha256:8241732fae22a7a36db7fd27a0fd1d8d52ec550240ca78a584a9fe5858c0d39e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QPDTWC7A4DHFMTBF3G2OGHPGIM/bundle.json","state_url":"https://pith.science/pith/QPDTWC7A4DHFMTBF3G2OGHPGIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QPDTWC7A4DHFMTBF3G2OGHPGIM/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":"2026-08-12T12:42:27Z","links":{"resolver":"https://pith.science/pith/QPDTWC7A4DHFMTBF3G2OGHPGIM","bundle":"https://pith.science/pith/QPDTWC7A4DHFMTBF3G2OGHPGIM/bundle.json","state":"https://pith.science/pith/QPDTWC7A4DHFMTBF3G2OGHPGIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QPDTWC7A4DHFMTBF3G2OGHPGIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QPDTWC7A4DHFMTBF3G2OGHPGIM","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":"a9163815129559208d2e7e082f10b423f59f967c94b4d8156337111b624283c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-24T07:30:03Z","title_canon_sha256":"f004434d35b0899dc8a81511215ba05c277ba5795977d79362c4286fa29488c5"},"schema_version":"1.0","source":{"id":"2605.24909","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.24909","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"arxiv_version","alias_value":"2605.24909v1","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.24909","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"pith_short_12","alias_value":"QPDTWC7A4DHF","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"pith_short_16","alias_value":"QPDTWC7A4DHFMTBF","created_at":"2026-05-26T01:04:05Z"},{"alias_kind":"pith_short_8","alias_value":"QPDTWC7A","created_at":"2026-05-26T01:04:05Z"}],"graph_snapshots":[{"event_id":"sha256:8241732fae22a7a36db7fd27a0fd1d8d52ec550240ca78a584a9fe5858c0d39e","target":"graph","created_at":"2026-05-26T01:04:05Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.24909/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $U_n=U_n(P,Q)$ be a nondegenerate Lucas sequence with $Q=\\pm 1$ and discriminant $\\Delta=P^2+4Q>0$. We study Diophantine equations \\[ A y^k=\\prod_{i=1}^r U_{n_i}(P,Q), \\qquad k\\geq 2, \\] where the indices $n_1,\\ldots,n_r$ are pairwise coprime. The strong divisibility property implies that the factors $U_{n_i}$ are pairwise coprime, and hence a global $k$-th power condition separates into local valuation conditions on the individual factors. For $k=2$, this gives a termwise square-class restriction: each $U_{n_i}$ has signed squarefree part supported on the primes dividing $A$. In particula","authors_text":"Dongyeon Kym","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-24T07:30:03Z","title":"Valuation Separation for Coprime Lucas Products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24909","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:b10ee1042dd36ed242923705c4ad349f073e83949f7e894da547bcd617dd09ae","target":"record","created_at":"2026-05-26T01:04:05Z","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":"a9163815129559208d2e7e082f10b423f59f967c94b4d8156337111b624283c5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-24T07:30:03Z","title_canon_sha256":"f004434d35b0899dc8a81511215ba05c277ba5795977d79362c4286fa29488c5"},"schema_version":"1.0","source":{"id":"2605.24909","kind":"arxiv","version":1}},"canonical_sha256":"83c73b0be0e0ce564c25d9b4e31de6433488c58e90434488e7b240bd45276d91","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"83c73b0be0e0ce564c25d9b4e31de6433488c58e90434488e7b240bd45276d91","first_computed_at":"2026-05-26T01:04:05.183393Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T01:04:05.183393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1GLT3nJvHe1Vww3tbOY3BnmcwB2kYXnEo38+o4JUMEXeENgttQGfqMR6pkMWFUqneIZBZBI6G7ePBMMX9ZqfAw==","signature_status":"signed_v1","signed_at":"2026-05-26T01:04:05.184177Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.24909","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b10ee1042dd36ed242923705c4ad349f073e83949f7e894da547bcd617dd09ae","sha256:8241732fae22a7a36db7fd27a0fd1d8d52ec550240ca78a584a9fe5858c0d39e"],"state_sha256":"04e6e012d074357cf1348bd7ebef5a8d83b001f9850a4d716ce82a3df9bd5c0b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c340yBCzR6MdD3ykyPae40pQ2kbQ+VFzItpIWRGYod3j7mE6wrRGccbD2MI4WYSubWk5gW0qkrwaZMHvapiFAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T12:42:27.459185Z","bundle_sha256":"dd4ebd8382d8f45b5f8326bba150f4008941da4a665196eccaa436d06919694a"}}