{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:LBMJY6TEW5V6QNUWY35WFSXQUQ","short_pith_number":"pith:LBMJY6TE","canonical_record":{"source":{"id":"2607.23662","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-26T13:53:01Z","cross_cats_sorted":[],"title_canon_sha256":"5d04594e345d4fc49b572813a7434344bd5cb9f4013af393a88b2e482fa422f5","abstract_canon_sha256":"e04c089423c3d71c370e2c206fb4000787f3fa80ae9681a8413eb39fcd0f9ab7"},"schema_version":"1.0"},"canonical_sha256":"58589c7a64b76be83696c6fb62caf0a4062a1e98db43a8bc24349eee7640e5d0","source":{"kind":"arxiv","id":"2607.23662","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.23662","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"arxiv_version","alias_value":"2607.23662v1","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.23662","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"pith_short_12","alias_value":"LBMJY6TEW5V6","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"pith_short_16","alias_value":"LBMJY6TEW5V6QNUW","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"pith_short_8","alias_value":"LBMJY6TE","created_at":"2026-07-28T01:23:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:LBMJY6TEW5V6QNUWY35WFSXQUQ","target":"record","payload":{"canonical_record":{"source":{"id":"2607.23662","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-26T13:53:01Z","cross_cats_sorted":[],"title_canon_sha256":"5d04594e345d4fc49b572813a7434344bd5cb9f4013af393a88b2e482fa422f5","abstract_canon_sha256":"e04c089423c3d71c370e2c206fb4000787f3fa80ae9681a8413eb39fcd0f9ab7"},"schema_version":"1.0"},"canonical_sha256":"58589c7a64b76be83696c6fb62caf0a4062a1e98db43a8bc24349eee7640e5d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T01:23:04.257570Z","signature_b64":"+eQyQVsgBXZ75aWeAVtzwBifrMO5Wp7a1cVtYtebV1TeMFu5z1vi6mti/++51OPj6jOW0tIPuG/FLpEY82LvDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58589c7a64b76be83696c6fb62caf0a4062a1e98db43a8bc24349eee7640e5d0","last_reissued_at":"2026-07-28T01:23:04.256791Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T01:23:04.256791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.23662","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":"2026-07-28T01:23:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XgoSdPSnyzB3eE8NXFKkub/abvxAytJ0o0z9vd7+M39flvNzLwaGHJw0vatPYwvKlbbImULi5ABVaNyjxkX8Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T22:31:24.843992Z"},"content_sha256":"f8da5a051952455071c2251a275fc9e4d7b6a8fc79e3809a34606eefe2b532bc","schema_version":"1.0","event_id":"sha256:f8da5a051952455071c2251a275fc9e4d7b6a8fc79e3809a34606eefe2b532bc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:LBMJY6TEW5V6QNUWY35WFSXQUQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Self-Referential Leading Digits of Exponential Sequences: Arithmetic Structure and Certified Search","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zihang Fang","submitted_at":"2026-07-26T13:53:01Z","abstract_excerpt":"For $c>1$ and an integer radix $b\\ge2$, we study the positive integers $m$ for which $mb^k\\le c^m<(m+1)b^k$ for some $k\\ge0$; for integer $c$, this is the self-prefix leading-digit condition. We derive an exact shrinking-target criterion; for $c\\ge2$, an exact signed-discrepancy identity isolates both infinitude and the conjectural logarithmic count. For $c\\ge2$ with nonintegral logarithmic slope, Lambert $W_{-1}$ inversion produces a candidate sequence with an eventual two-gap law and an exact counting formula; for $(c,b)=(2,10)$ all consecutive candidate gaps are $3$ or $4$. For algebraic $c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23662","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/2607.23662/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":"2026-07-28T01:23:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2zExuopXg8wBFvMs3XCPdt5oEYdXdHPnK2Uzrq+unv8v9rQf2UUyyhTdgWQQJHxoBQkwhEcaD+aWz8zuZdvACA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T22:31:24.844547Z"},"content_sha256":"effffea34f5130f134a9926b541ca229d55fa5be5d68a66149643e1e249fd2c1","schema_version":"1.0","event_id":"sha256:effffea34f5130f134a9926b541ca229d55fa5be5d68a66149643e1e249fd2c1"},{"event_type":"integrity_finding","subject_pith_number":"pith:2026:LBMJY6TEW5V6QNUWY35WFSXQUQ","target":"integrity","payload":{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. A longer candidate (10.1070/IM2000v064n06ABEH000314) was visible in the surrounding text but could not be confirmed against doi.org as printed.","snippet":"E. M. Matveev. “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II”. In:Izvestiya: Mathematics64.6 (2000), pp. 1217–1269.doi:10.1070/IM2000v064n06ABEH 000314","arxiv_id":"2607.23662","detector":"doi_compliance","evidence":{"ref_index":22,"verdict_class":"incontrovertible","resolved_title":null,"printed_excerpt":"10.1070/im2000v064n06abeh","reconstructed_doi":"10.1070/IM2000v064n06ABEH000314"},"severity":"advisory","ref_index":22,"audited_at":"2026-07-30T19:21:48.345863Z","event_type":"pith.integrity.v1","detected_doi":"10.1070/IM2000v064n06ABEH000314","detector_url":"https://pith.science/pith-integrity-protocol#doi_compliance","external_url":null,"finding_type":"recoverable_identifier","evidence_hash":"2c6aadabc2e7e3b91de2c8d90d729c4f7915211f72edc5a63aa6f64c4243129d","paper_version":1,"verdict_class":"incontrovertible","resolved_title":null,"detector_version":"1.1.0","detected_arxiv_id":null,"integrity_event_id":14038,"payload_sha256":"aa65a963625b21a16482ab998215434085fb0d571d865a2edfed1e8744ed8560","signature_b64":null,"signing_key_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-30T19:26:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TPFAmP44rkrzXoJvn8R3yfkeYhHqdjm+xnYZU4e6InM1MIQeAiCXb82eUW7BuT9kVnCxJYTPQA7JoSSDGS+KAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T22:31:24.848708Z"},"content_sha256":"21278da055c4c58fee9cdb8706023e280c6fcc86b74c0129cdaf33fdbd687320","schema_version":"1.0","event_id":"sha256:21278da055c4c58fee9cdb8706023e280c6fcc86b74c0129cdaf33fdbd687320"},{"event_type":"integrity_finding","subject_pith_number":"pith:2026:LBMJY6TEW5V6QNUWY35WFSXQUQ","target":"integrity","payload":{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. A longer candidate (10.1017/CBO9780511921698) was visible in the surrounding text but could not be confirmed against doi.org as printed.","snippet":"R. P. Brent and P. Zimmermann.Modern Computer Arithmetic. Cambridge Monographs on Applied and Computational Mathematics 18. Cambridge: Cambridge University Press, 2010.doi:10.1017/CBO978051 1921698","arxiv_id":"2607.23662","detector":"doi_compliance","evidence":{"ref_index":4,"verdict_class":"incontrovertible","resolved_title":null,"printed_excerpt":"10.1017/cbo978051","reconstructed_doi":"10.1017/CBO9780511921698"},"severity":"advisory","ref_index":4,"audited_at":"2026-07-30T19:21:48.345863Z","event_type":"pith.integrity.v1","detected_doi":"10.1017/CBO9780511921698","detector_url":"https://pith.science/pith-integrity-protocol#doi_compliance","external_url":null,"finding_type":"recoverable_identifier","evidence_hash":"b4669eaa5e535514d9fa20047b4918202b688696f859b06e4cf46af2d02f5193","paper_version":1,"verdict_class":"incontrovertible","resolved_title":null,"detector_version":"1.1.0","detected_arxiv_id":null,"integrity_event_id":14037,"payload_sha256":"99a9f37c73893a939c0ab93f357638affcb0615623795cf49c9f0bdc7e0a7878","signature_b64":null,"signing_key_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-30T19:26:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"olwilH7Eh9JAoChTiYMegbUMhlqL9qv3u5tD7mnA9hs4JipzlUxoW8O37M9Zwy+1MauwdZfyA+0mXHtAI3+OAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T22:31:24.849218Z"},"content_sha256":"072577403c5d6cfecd8a4799933453aae7fcbe059be1fce7bc8037bce9d47ec2","schema_version":"1.0","event_id":"sha256:072577403c5d6cfecd8a4799933453aae7fcbe059be1fce7bc8037bce9d47ec2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LBMJY6TEW5V6QNUWY35WFSXQUQ/bundle.json","state_url":"https://pith.science/pith/LBMJY6TEW5V6QNUWY35WFSXQUQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LBMJY6TEW5V6QNUWY35WFSXQUQ/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":"2026-08-20T22:31:24Z","links":{"resolver":"https://pith.science/pith/LBMJY6TEW5V6QNUWY35WFSXQUQ","bundle":"https://pith.science/pith/LBMJY6TEW5V6QNUWY35WFSXQUQ/bundle.json","state":"https://pith.science/pith/LBMJY6TEW5V6QNUWY35WFSXQUQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LBMJY6TEW5V6QNUWY35WFSXQUQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:LBMJY6TEW5V6QNUWY35WFSXQUQ","merge_version":"pith-open-graph-merge-v1","event_count":4,"valid_event_count":4,"invalid_event_count":0,"equivocation_count":1,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e04c089423c3d71c370e2c206fb4000787f3fa80ae9681a8413eb39fcd0f9ab7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-26T13:53:01Z","title_canon_sha256":"5d04594e345d4fc49b572813a7434344bd5cb9f4013af393a88b2e482fa422f5"},"schema_version":"1.0","source":{"id":"2607.23662","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.23662","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"arxiv_version","alias_value":"2607.23662v1","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.23662","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"pith_short_12","alias_value":"LBMJY6TEW5V6","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"pith_short_16","alias_value":"LBMJY6TEW5V6QNUW","created_at":"2026-07-28T01:23:04Z"},{"alias_kind":"pith_short_8","alias_value":"LBMJY6TE","created_at":"2026-07-28T01:23:04Z"}],"graph_snapshots":[{"event_id":"sha256:effffea34f5130f134a9926b541ca229d55fa5be5d68a66149643e1e249fd2c1","target":"graph","created_at":"2026-07-28T01:23:04Z","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/2607.23662/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For $c>1$ and an integer radix $b\\ge2$, we study the positive integers $m$ for which $mb^k\\le c^m<(m+1)b^k$ for some $k\\ge0$; for integer $c$, this is the self-prefix leading-digit condition. We derive an exact shrinking-target criterion; for $c\\ge2$, an exact signed-discrepancy identity isolates both infinitude and the conjectural logarithmic count. For $c\\ge2$ with nonintegral logarithmic slope, Lambert $W_{-1}$ inversion produces a candidate sequence with an eventual two-gap law and an exact counting formula; for $(c,b)=(2,10)$ all consecutive candidate gaps are $3$ or $4$. For algebraic $c","authors_text":"Zihang Fang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-26T13:53:01Z","title":"Self-Referential Leading Digits of Exponential Sequences: Arithmetic Structure and Certified Search"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23662","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:f8da5a051952455071c2251a275fc9e4d7b6a8fc79e3809a34606eefe2b532bc","target":"record","created_at":"2026-07-28T01:23:04Z","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":"e04c089423c3d71c370e2c206fb4000787f3fa80ae9681a8413eb39fcd0f9ab7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-26T13:53:01Z","title_canon_sha256":"5d04594e345d4fc49b572813a7434344bd5cb9f4013af393a88b2e482fa422f5"},"schema_version":"1.0","source":{"id":"2607.23662","kind":"arxiv","version":1}},"canonical_sha256":"58589c7a64b76be83696c6fb62caf0a4062a1e98db43a8bc24349eee7640e5d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58589c7a64b76be83696c6fb62caf0a4062a1e98db43a8bc24349eee7640e5d0","first_computed_at":"2026-07-28T01:23:04.256791Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-28T01:23:04.256791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+eQyQVsgBXZ75aWeAVtzwBifrMO5Wp7a1cVtYtebV1TeMFu5z1vi6mti/++51OPj6jOW0tIPuG/FLpEY82LvDg==","signature_status":"signed_v1","signed_at":"2026-07-28T01:23:04.257570Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.23662","source_kind":"arxiv","source_version":1}}},"equivocations":[{"signer_id":"pith.science","event_type":"integrity_finding","target":"integrity","event_ids":["sha256:072577403c5d6cfecd8a4799933453aae7fcbe059be1fce7bc8037bce9d47ec2","sha256:21278da055c4c58fee9cdb8706023e280c6fcc86b74c0129cdaf33fdbd687320"]}],"invalid_events":[],"applied_event_ids":["sha256:f8da5a051952455071c2251a275fc9e4d7b6a8fc79e3809a34606eefe2b532bc","sha256:effffea34f5130f134a9926b541ca229d55fa5be5d68a66149643e1e249fd2c1"],"state_sha256":"2b5e287724a67734b2af1475226a9414ed9bdd6cf93ab0bbb4ccc3d33597daab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jSHrCAtxq8oqdNvNqn/1OC/+NCwrRxNR5mxf68E+fLFQ1gUJDHoihia0WA4LKlRZk/XUnq50EIx6HRTypD/oCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T22:31:24.851730Z","bundle_sha256":"49b5f0934f07189d6337faa49cd36e168ef155ff03227cd42ba2d1eac799b987"}}