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We prove unconditionally that Proposition $(1+1.9)$ is true. Assuming the Elliott--Halberstam Conjecture, the exponent $1.9$ can be improved to $1.4$. Analogously, Proposition $(1-a)$ is formulated for the Twin Prime Conjecture. Unconditionally, we prove Proposition $(1-1"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.05224","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-01T15:03:44Z","cross_cats_sorted":[],"title_canon_sha256":"23a9b31d37fd8b6d5ca4c8e08e178876e0cdd5b3dfc6327c0d63ca81620ad9e3","abstract_canon_sha256":"c2725b59f6035228be9d5ee6a37813ba59c72249e504bc28ae86b6301e0d6de6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T00:13:49.283228Z","signature_b64":"bi9XsoHRvDoMqDSNO2VuAMZgrNvBcihARybW3+APwGloG4aQPtrYjzNxOmXY/4nLiHkU1phvGLdfUfBssDPdAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ced3b8c2bf3b004dc75fd26364ff2e4638a543c82076a0253bc87752a3e24c4","last_reissued_at":"2026-06-05T00:13:49.282542Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T00:13:49.282542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Theorem $(1+1.9)$ on the Goldbach Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jiamin Li, Jianya Liu","submitted_at":"2026-06-01T15:03:44Z","abstract_excerpt":"For $1 \\leq a \\leq 2$, we say Proposition $(1+a)$ holds if every sufficiently large even integer $N$ can be written as $$N = p + rq, \\quad r \\leq q^{a-1},$$ where $r$ is either $1$ or prime, and $p,q$ are primes. Thus Proposition $(1+1)$ is essentially the binary Goldbach Conjecture, and Proposition $(1+2)$ is Chen's theorem. We prove unconditionally that Proposition $(1+1.9)$ is true. Assuming the Elliott--Halberstam Conjecture, the exponent $1.9$ can be improved to $1.4$. Analogously, Proposition $(1-a)$ is formulated for the Twin Prime Conjecture. Unconditionally, we prove Proposition $(1-1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05224","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/2606.05224/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.05224","created_at":"2026-06-05T00:13:49.282651+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.05224v1","created_at":"2026-06-05T00:13:49.282651+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05224","created_at":"2026-06-05T00:13:49.282651+00:00"},{"alias_kind":"pith_short_12","alias_value":"LTWTXDBL6OYA","created_at":"2026-06-05T00:13:49.282651+00:00"},{"alias_kind":"pith_short_16","alias_value":"LTWTXDBL6OYAJXDV","created_at":"2026-06-05T00:13:49.282651+00:00"},{"alias_kind":"pith_short_8","alias_value":"LTWTXDBL","created_at":"2026-06-05T00:13:49.282651+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R","json":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R.json","graph_json":"https://pith.science/api/pith-number/LTWTXDBL6OYAJXDV7UTDMT7S4R/graph.json","events_json":"https://pith.science/api/pith-number/LTWTXDBL6OYAJXDV7UTDMT7S4R/events.json","paper":"https://pith.science/paper/LTWTXDBL"},"agent_actions":{"view_html":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R","download_json":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R.json","view_paper":"https://pith.science/paper/LTWTXDBL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.05224&json=true","fetch_graph":"https://pith.science/api/pith-number/LTWTXDBL6OYAJXDV7UTDMT7S4R/graph.json","fetch_events":"https://pith.science/api/pith-number/LTWTXDBL6OYAJXDV7UTDMT7S4R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R/action/storage_attestation","attest_author":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R/action/author_attestation","sign_citation":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R/action/citation_signature","submit_replication":"https://pith.science/pith/LTWTXDBL6OYAJXDV7UTDMT7S4R/action/replication_record"}},"created_at":"2026-06-05T00:13:49.282651+00:00","updated_at":"2026-06-05T00:13:49.282651+00:00"}