{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ZA6LU6VQHTNHGLTBJ7NWDFRUSR","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":"bc90839d14e0dc4faa41b14a1fd2c34aed2bc6d5e091437704dd6644b2ba9b11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-24T13:59:53Z","title_canon_sha256":"c2c2ae39f3771848ea068cbb74a0e04135d4e2eb80173adfcc132dd9b3ad1caf"},"schema_version":"1.0","source":{"id":"2606.25849","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.25849","created_at":"2026-06-25T01:18:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.25849v1","created_at":"2026-06-25T01:18:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.25849","created_at":"2026-06-25T01:18:41Z"},{"alias_kind":"pith_short_12","alias_value":"ZA6LU6VQHTNH","created_at":"2026-06-25T01:18:41Z"},{"alias_kind":"pith_short_16","alias_value":"ZA6LU6VQHTNHGLTB","created_at":"2026-06-25T01:18:41Z"},{"alias_kind":"pith_short_8","alias_value":"ZA6LU6VQ","created_at":"2026-06-25T01:18:41Z"}],"graph_snapshots":[{"event_id":"sha256:82e14d0882719a91a3c07b836b91c3951cd7b6c8e4d5b79686e0b49e295cb0d8","target":"graph","created_at":"2026-06-25T01:18:41Z","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/2606.25849/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We resolve Erd\\H{o}s Problem 1061, the question whether the number \\[\n  S(x)=\\#\\{(a,b)\\in\\mathbb{N}^2:a+b\\le x,\n  \\ \\sigma(a)+\\sigma(b)=\\sigma(a+b)\\} \\] of ordered solutions has a linear asymptotic $S(x)\\sim cx$. In fact the opposite extreme holds at every fixed logarithmic scale: for every \\(R>0\\), \\[\n  \\lim_{x\\to\\infty}\\frac{S(x)}{x(\\log x)^R}=+\\infty. \\] The construction begins with three integers having the same abundancy index and reduces the divisor-sum identity to two equations in six primes. After a linear change of variables, these equations lie on a split quadric. A three-parameter r","authors_text":"Eric Li (Trinity College, University of Cambridge)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-24T13:59:53Z","title":"A resolution of Erd\\H{o}s Problem 1061 on the sum-of-divisors function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25849","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:c74b06884737c55a93eae2f3bf4275f847b8b711b31bf669a97dcf538d64b59a","target":"record","created_at":"2026-06-25T01:18:41Z","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":"bc90839d14e0dc4faa41b14a1fd2c34aed2bc6d5e091437704dd6644b2ba9b11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-24T13:59:53Z","title_canon_sha256":"c2c2ae39f3771848ea068cbb74a0e04135d4e2eb80173adfcc132dd9b3ad1caf"},"schema_version":"1.0","source":{"id":"2606.25849","kind":"arxiv","version":1}},"canonical_sha256":"c83cba7ab03cda732e614fdb6196349444761af6d4fde208d484e5f77779d341","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c83cba7ab03cda732e614fdb6196349444761af6d4fde208d484e5f77779d341","first_computed_at":"2026-06-25T01:18:41.048894Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-25T01:18:41.048894Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EsIua3efK2uM4tzvTnDcboTSUd+c/EkfEC91+4xeWCahBqJDbpbPb50Jn8kS+EV+A49OaGZnrGejU9AmJUnHAA==","signature_status":"signed_v1","signed_at":"2026-06-25T01:18:41.049259Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.25849","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c74b06884737c55a93eae2f3bf4275f847b8b711b31bf669a97dcf538d64b59a","sha256:82e14d0882719a91a3c07b836b91c3951cd7b6c8e4d5b79686e0b49e295cb0d8"],"state_sha256":"29dbcbd6de2a948e5282da15766616108762509ea8cf33a54610992b68343e1e"}