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We obtain sim"},"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":"2605.28781","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-27T17:42:41Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"33202877d5e42539d39d14da42b302bc88e4f5a49f02dee39e3e2a5559c8c707","abstract_canon_sha256":"9c563951d628ecb86e163eb936c076c1058c069a1721989a940343292ca88e49"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T02:05:03.466273Z","signature_b64":"uGSA0PpcOa4Byu8FcT8em/qRw4SXPKq14z0/BRqKwodv4bDN5twrHiFBkYYKqM4ZygXbYeradSdl4jiGKq9ZBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e04bab4a72f6a2792d147cac7b8c238a4fe9d1b0c5200b07dadad3bcb06b835","last_reissued_at":"2026-05-28T02:05:03.465874Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T02:05:03.465874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The sum-product conjecture is false for real numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Carl Schildkraut, Dmitrii Zhelezov, Thomas F Bloom, Will Sawin","submitted_at":"2026-05-27T17:42:41Z","abstract_excerpt":"We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\\subset \\mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\\asymp \\log\\lvert A\\rvert$) such that \\[\\max(\\lvert A+A\\rvert ,\\lvert AA\\rvert)\\leq \\lvert A\\rvert^{2-c}\\] where $c>0$ is an absolute constant.\n  We also disprove the many sums and products conjecture by constructing, for any $k\\geq 3$, arbitrarily large $A\\subset \\mathbb{R}$ such that \\[\\max(\\lvert kA\\rvert,\\lvert A^{(k)}\\rvert)\\leq \\lvert A\\rvert^{C\\frac{\\log k}{\\log\\log k}}\\] for some constant $C>0$. We obtain sim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28781","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/2605.28781/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":"2605.28781","created_at":"2026-05-28T02:05:03.465926+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.28781v1","created_at":"2026-05-28T02:05:03.465926+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.28781","created_at":"2026-05-28T02:05:03.465926+00:00"},{"alias_kind":"pith_short_12","alias_value":"BYCLVNFHF5VC","created_at":"2026-05-28T02:05:03.465926+00:00"},{"alias_kind":"pith_short_16","alias_value":"BYCLVNFHF5VCPEWR","created_at":"2026-05-28T02:05:03.465926+00:00"},{"alias_kind":"pith_short_8","alias_value":"BYCLVNFH","created_at":"2026-05-28T02:05:03.465926+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/BYCLVNFHF5VCPEWRI7FMPOGCHC","json":"https://pith.science/pith/BYCLVNFHF5VCPEWRI7FMPOGCHC.json","graph_json":"https://pith.science/api/pith-number/BYCLVNFHF5VCPEWRI7FMPOGCHC/graph.json","events_json":"https://pith.science/api/pith-number/BYCLVNFHF5VCPEWRI7FMPOGCHC/events.json","paper":"https://pith.science/paper/BYCLVNFH"},"agent_actions":{"view_html":"https://pith.science/pith/BYCLVNFHF5VCPEWRI7FMPOGCHC","download_json":"https://pith.science/pith/BYCLVNFHF5VCPEWRI7FMPOGCHC.json","view_paper":"https://pith.science/paper/BYCLVNFH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.28781&json=true","fetch_graph":"https://pith.science/api/pith-number/BYCLVNFHF5VCPEWRI7FMPOGCHC/graph.json","fetch_events":"https://pith.science/api/pith-number/BYCLVNFHF5VCPEWRI7FMPOGCHC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BYCLVNFHF5VCPEWRI7FMPOGCHC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BYCLVNFHF5VCPEWRI7FMPOGCHC/action/storage_attestation","attest_author":"https://pith.science/pith/BYCLVNFHF5VCPEWRI7FMPOGCHC/action/author_attestation","sign_citation":"https://pith.science/pith/BYCLVNFHF5VCPEWRI7FMPOGCHC/action/citation_signature","submit_replication":"https://pith.science/pith/BYCLVNFHF5VCPEWRI7FMPOGCHC/action/replication_record"}},"created_at":"2026-05-28T02:05:03.465926+00:00","updated_at":"2026-05-28T02:05:03.465926+00:00"}