{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7I2CSU33OAZKJB5KEIZ6KC4YJ3","short_pith_number":"pith:7I2CSU33","canonical_record":{"source":{"id":"2607.28857","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-30T21:44:54Z","cross_cats_sorted":[],"title_canon_sha256":"ccd9a3131d8e51d36f0a40dd9c408d726b2fed01ea7ea3c4de155404e660414e","abstract_canon_sha256":"efce3effb53b4ade4ca9dc5c3ca51d88116e5a482ad4a7155ece63c2f1b806d5"},"schema_version":"1.0"},"canonical_sha256":"fa3429537b7032a487aa2233e50b984ed1ec23218b07fed36024abecf1d43ef6","source":{"kind":"arxiv","id":"2607.28857","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.28857","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"arxiv_version","alias_value":"2607.28857v1","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.28857","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"pith_short_12","alias_value":"7I2CSU33OAZK","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"pith_short_16","alias_value":"7I2CSU33OAZKJB5K","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"pith_short_8","alias_value":"7I2CSU33","created_at":"2026-08-03T01:17:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7I2CSU33OAZKJB5KEIZ6KC4YJ3","target":"record","payload":{"canonical_record":{"source":{"id":"2607.28857","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-30T21:44:54Z","cross_cats_sorted":[],"title_canon_sha256":"ccd9a3131d8e51d36f0a40dd9c408d726b2fed01ea7ea3c4de155404e660414e","abstract_canon_sha256":"efce3effb53b4ade4ca9dc5c3ca51d88116e5a482ad4a7155ece63c2f1b806d5"},"schema_version":"1.0"},"canonical_sha256":"fa3429537b7032a487aa2233e50b984ed1ec23218b07fed36024abecf1d43ef6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-03T01:17:13.453964Z","signature_b64":"82YqekMRwyuTMVemf8lB/44fl3LExO81rgsALcKdOOX9pMYeQQlNHiEFdWlbakWJ6iIvS8r+IVgrislZXghhBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa3429537b7032a487aa2233e50b984ed1ec23218b07fed36024abecf1d43ef6","last_reissued_at":"2026-08-03T01:17:13.451949Z","signature_status":"signed_v1","first_computed_at":"2026-08-03T01:17:13.451949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.28857","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":"2026-08-03T01:17:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vb8ffLfYCn6TH5LmEmgr+TcCYmEOP+LsnvNVFme4sZqCsl+3YPSNVg4hUYwhDIfGIJS9gCgkCrIm1zhRmXXlCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T12:40:47.462043Z"},"content_sha256":"5a4f8087c3aa8fbce08cb7f06c7d3e040de248c81b87f300559ca671a8c3991b","schema_version":"1.0","event_id":"sha256:5a4f8087c3aa8fbce08cb7f06c7d3e040de248c81b87f300559ca671a8c3991b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7I2CSU33OAZKJB5KEIZ6KC4YJ3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiplicative dependence in the sumset of multiplicative groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, Yuri Bilu","submitted_at":"2026-07-30T21:44:54Z","abstract_excerpt":"Let $\\Gamma$ and $\\Delta$ be finitely generated multiplicative groups of algebraic numbers such that $\\Gamma\\cap\\Delta$ is a finite group. We show that, up to finitely many exceptions, non-zero sums $x_1+y_1$ and $x_2+y_2$, with $x_1, x_2\\in \\Gamma$ and $y_1,y_2\\in \\Delta$, are multiplicatively dependent only if $x_1/x_2=y_1/y_2$ is a root of unity. For $m\\ge 3$, we discuss possible shapes of $m$ multiplicatively dependent sums $x_1+y_1, \\ \\ldots, \\ x_m+y_m$ with $x_1, \\ldots, x_m \\in \\Gamma$ and $y_1, \\ldots, y_m \\in \\Delta$. For $m=3$ we classify such sums, up to finitely many exceptions, as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28857","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/2607.28857/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":"2026-08-03T01:17:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G8lNQ8gahXJAQJ55fhqXb5oBqLko8rbhdN4/2EVdyNIUDRvvs1/6gPz95GXCvFzqtMFbJ4GOPPxvZM5GN/S3AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T12:40:47.462590Z"},"content_sha256":"c0d562cfe8a12c05f76b2818aebd57f70c99dacc851b652542c55dbe39ebcef3","schema_version":"1.0","event_id":"sha256:c0d562cfe8a12c05f76b2818aebd57f70c99dacc851b652542c55dbe39ebcef3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/bundle.json","state_url":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/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":"2026-08-12T12:40:47Z","links":{"resolver":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3","bundle":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/bundle.json","state":"https://pith.science/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7I2CSU33OAZKJB5KEIZ6KC4YJ3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7I2CSU33OAZKJB5KEIZ6KC4YJ3","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":"efce3effb53b4ade4ca9dc5c3ca51d88116e5a482ad4a7155ece63c2f1b806d5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-30T21:44:54Z","title_canon_sha256":"ccd9a3131d8e51d36f0a40dd9c408d726b2fed01ea7ea3c4de155404e660414e"},"schema_version":"1.0","source":{"id":"2607.28857","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.28857","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"arxiv_version","alias_value":"2607.28857v1","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.28857","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"pith_short_12","alias_value":"7I2CSU33OAZK","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"pith_short_16","alias_value":"7I2CSU33OAZKJB5K","created_at":"2026-08-03T01:17:13Z"},{"alias_kind":"pith_short_8","alias_value":"7I2CSU33","created_at":"2026-08-03T01:17:13Z"}],"graph_snapshots":[{"event_id":"sha256:c0d562cfe8a12c05f76b2818aebd57f70c99dacc851b652542c55dbe39ebcef3","target":"graph","created_at":"2026-08-03T01:17:13Z","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/2607.28857/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\Gamma$ and $\\Delta$ be finitely generated multiplicative groups of algebraic numbers such that $\\Gamma\\cap\\Delta$ is a finite group. We show that, up to finitely many exceptions, non-zero sums $x_1+y_1$ and $x_2+y_2$, with $x_1, x_2\\in \\Gamma$ and $y_1,y_2\\in \\Delta$, are multiplicatively dependent only if $x_1/x_2=y_1/y_2$ is a root of unity. For $m\\ge 3$, we discuss possible shapes of $m$ multiplicatively dependent sums $x_1+y_1, \\ \\ldots, \\ x_m+y_m$ with $x_1, \\ldots, x_m \\in \\Gamma$ and $y_1, \\ldots, y_m \\in \\Delta$. For $m=3$ we classify such sums, up to finitely many exceptions, as","authors_text":"Florian Luca, Yuri Bilu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-30T21:44:54Z","title":"Multiplicative dependence in the sumset of multiplicative groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28857","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":"sha256:5a4f8087c3aa8fbce08cb7f06c7d3e040de248c81b87f300559ca671a8c3991b","target":"record","created_at":"2026-08-03T01:17:13Z","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":"efce3effb53b4ade4ca9dc5c3ca51d88116e5a482ad4a7155ece63c2f1b806d5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-30T21:44:54Z","title_canon_sha256":"ccd9a3131d8e51d36f0a40dd9c408d726b2fed01ea7ea3c4de155404e660414e"},"schema_version":"1.0","source":{"id":"2607.28857","kind":"arxiv","version":1}},"canonical_sha256":"fa3429537b7032a487aa2233e50b984ed1ec23218b07fed36024abecf1d43ef6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa3429537b7032a487aa2233e50b984ed1ec23218b07fed36024abecf1d43ef6","first_computed_at":"2026-08-03T01:17:13.451949Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-03T01:17:13.451949Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"82YqekMRwyuTMVemf8lB/44fl3LExO81rgsALcKdOOX9pMYeQQlNHiEFdWlbakWJ6iIvS8r+IVgrislZXghhBA==","signature_status":"signed_v1","signed_at":"2026-08-03T01:17:13.453964Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.28857","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a4f8087c3aa8fbce08cb7f06c7d3e040de248c81b87f300559ca671a8c3991b","sha256:c0d562cfe8a12c05f76b2818aebd57f70c99dacc851b652542c55dbe39ebcef3"],"state_sha256":"e0162a85473cff4e7e90e3f63beb7c626aba4c95ef6bde478dd556cdc7a6da79"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8GtLZoCHzYqh6AAGLlBTMCVyPsETWXNdbqzNzGS7tNu/iJ758Ncv9Zf3YvGXyAYJzV5RqlZdzqoxjDDurZVsAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T12:40:47.467651Z","bundle_sha256":"3030ba7a7194cda59fd7e6de3f61c114bdf8dfe795b3ec667cd3855e3bf64a3f"}}