{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:SFGFD7GNZZROH42YHPDSIGHKXV","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":"fc61098449ce1a8bf876281e54072f7e812ca1f5e7fa799aca163cac4b1eb698","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-04T02:31:00Z","title_canon_sha256":"3ad06a4ba6d21ec394320ff6a5665f5852ae710177298d2e1024f7ea81eb49a0"},"schema_version":"1.0","source":{"id":"2607.03662","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.03662","created_at":"2026-07-07T02:17:59Z"},{"alias_kind":"arxiv_version","alias_value":"2607.03662v1","created_at":"2026-07-07T02:17:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.03662","created_at":"2026-07-07T02:17:59Z"},{"alias_kind":"pith_short_12","alias_value":"SFGFD7GNZZRO","created_at":"2026-07-07T02:17:59Z"},{"alias_kind":"pith_short_16","alias_value":"SFGFD7GNZZROH42Y","created_at":"2026-07-07T02:17:59Z"},{"alias_kind":"pith_short_8","alias_value":"SFGFD7GN","created_at":"2026-07-07T02:17:59Z"}],"graph_snapshots":[{"event_id":"sha256:4c5599c3f40f911c49dd4438ea0f8a36f054f5425e63b181b0d00b2aa93509fa","target":"graph","created_at":"2026-07-07T02:17:59Z","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/2607.03662/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\\ge$ 1:$\\Omega(m)$ $\\le$ 2}. We prove that the sumset $P_2$+{$a^a$: a$\\ge$ 1} has positive lower density. The proof uses the Romanoff second moment method, in the spirit of Li and Pan's theorem on $P_2$+$2^{\\mathcal P}$. The main new ingredient is the following average estimate for the singular factor \\[\n  \\frac{1}{K(K-1)}\n  \\sum_{\\substack{1\\le a,b\\le K\\\\a\\ne b}}\n  \\prod_{p\\mid a^a-b^b}\\left(1+\\frac{\\kappa}{p}\\right)\n  \\le C_\\kappa \\] for some constant $C_\\kappa>0$, which is valid for all","authors_text":"Huixi Li, Junfeng Li, Yuchen Ding","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-04T02:31:00Z","title":"A Romanoff-type theorem for $P_2$+{$a^a$: a$\\ge$ 1}"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03662","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":"sha256:6c631c266ee2f08ad8c61d1c3120001f977ba81ad3838c28b1922a93fb55da81","target":"record","created_at":"2026-07-07T02:17:59Z","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":"fc61098449ce1a8bf876281e54072f7e812ca1f5e7fa799aca163cac4b1eb698","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-04T02:31:00Z","title_canon_sha256":"3ad06a4ba6d21ec394320ff6a5665f5852ae710177298d2e1024f7ea81eb49a0"},"schema_version":"1.0","source":{"id":"2607.03662","kind":"arxiv","version":1}},"canonical_sha256":"914c51fccdce62e3f3583bc72418eabd6aaaab2307844eccffc10c7e6caa9873","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"914c51fccdce62e3f3583bc72418eabd6aaaab2307844eccffc10c7e6caa9873","first_computed_at":"2026-07-07T02:17:59.960615Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:17:59.960615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CSRFS8fBaY0xEXtYLf2u3Hk88YrQwpkH3BJFIlmr3ST2ifx0Afawul3dubrcIcB5di7fCD7C5qjhVRcbMEjnDw==","signature_status":"signed_v1","signed_at":"2026-07-07T02:17:59.961414Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.03662","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c631c266ee2f08ad8c61d1c3120001f977ba81ad3838c28b1922a93fb55da81","sha256:4c5599c3f40f911c49dd4438ea0f8a36f054f5425e63b181b0d00b2aa93509fa"],"state_sha256":"9e1bae6c2a6f68b61f4f5ac56fdcb19fa2a644c6369c25ec9ea44961c75a4433"}