{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2AAHHZ2E4A4MKFRPTRP7RG6PN5","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":"923c86fc0e7aea428b4c9c9477cbaeea5b08e23bd608e2c6ecfeeecc3081db03","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T21:26:51Z","title_canon_sha256":"7977e62c51996c1caf8eec07fbc57eccfc08fb03696928e4680bc6257c4017da"},"schema_version":"1.0","source":{"id":"2605.15434","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.15434","created_at":"2026-05-20T00:00:58Z"},{"alias_kind":"arxiv_version","alias_value":"2605.15434v1","created_at":"2026-05-20T00:00:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.15434","created_at":"2026-05-20T00:00:58Z"},{"alias_kind":"pith_short_12","alias_value":"2AAHHZ2E4A4M","created_at":"2026-05-20T00:00:58Z"},{"alias_kind":"pith_short_16","alias_value":"2AAHHZ2E4A4MKFRP","created_at":"2026-05-20T00:00:58Z"},{"alias_kind":"pith_short_8","alias_value":"2AAHHZ2E","created_at":"2026-05-20T00:00:58Z"}],"graph_snapshots":[{"event_id":"sha256:c948545b55151929762d9b464aa4efd98104d42f080ebebfb7197e9b5f254368","target":"graph","created_at":"2026-05-20T00:00:58Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"When h = N^2 + O(N), the number of solutions admits an asymptotic formula with square-root cancellation error terms obtained by exploiting symmetry via Ramanujan sums and bypassing Kloosterman sum bounds."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The additional symmetry present precisely when h = N^2 + O(N) permits direct use of Ramanujan sums to achieve square-root cancellation without relying on general Kloosterman bounds."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Proves asymptotic count of solutions to x1 x2 - x3 x4 = h for xi in [-N, N] with square-root cancellation when h = N^2 + O(N), confirming a prior speculation."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"When h equals N squared plus O(N), the number of solutions to x1 x2 minus x3 x4 equals h inside the box of side 2N admits an asymptotic with square-root cancellation error terms."}],"snapshot_sha256":"65f5b80df6ff5052d89ea8b2264b67de2574bb67015bc82d172e7f2df7394d8d"},"formal_canon":{"evidence_count":1,"snapshot_sha256":"a9e79524272a57ea32c64691261ec703b8798fa81e12c2107d2168b18316e877"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"cited_work_retraction","ran_at":"2026-05-19T15:54:28.079071Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"citation_quote_validity","ran_at":"2026-05-19T15:50:29.980697Z","status":"completed","version":"0.1.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T15:01:38.358032Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T15:01:17.677136Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.127326Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.692019Z","status":"skipped","version":"1.0.0"}],"endpoint":"/pith/2605.15434/integrity.json","findings":[],"snapshot_sha256":"52c17a2e40dbb3e1ce80b10560a6066a9d2639ad6d05ae6c183156f87c061f3b","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given $h, N \\in \\mathbb{N}$ satisfying $1 \\leqslant h \\leqslant N^2$, we prove an asymptotic formula for the number of solutions to the equation $x_1 x_2 - x_3 x_4 = h$ with $x_1, \\ldots, x_4 \\in [-N,N] \\cap \\mathbb{Z}$. We use a combination of combinatorial and analytic arguments in physical space along with bounds for Kloosterman sums. Our main result concerns the case when $h = N^2 + O(N)$, wherein we obtain square-root cancellation error terms by bypassing Kloosterman sum bounds and exploiting an additional symmetry available in this setting via Ramanujan sums. This confirms a speculation ","authors_text":"Akshat Mudgal, Jonathan Chapman","cross_cats":[],"headline":"When h equals N squared plus O(N), the number of solutions to x1 x2 minus x3 x4 equals h inside the box of side 2N admits an asymptotic with square-root cancellation error terms.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T21:26:51Z","title":"Counting solutions to the quadratic determinant equation"},"references":{"count":18,"internal_anchors":0,"resolved_work":18,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"M. Afifurrahman,A uniform formula on the number of integer matrices with given determinant and height, J. Number Theory281(2026), 741–770","work_id":"32676eb2-ef2d-4bb2-8c0e-1df3a59f738b","year":2026},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Apostol,Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer- Verlag, New York-Heidelberg, 1976","work_id":"7028696b-1644-4236-b71f-c50128b5412e","year":1976},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"J. Chapman, A. Mudgal,On commuting integer matrices, arXiv:2504.15839, to appear in Trans. Amer. Math. Soc","work_id":"dd70991f-c80b-4726-95e7-26ab68e4d652","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"J. Chapman, A. Mudgal,Counting2×2integer matrices with a given determinant, arXiv:2509.20259","work_id":"776f1133-263c-441f-bd17-ae32c54f0703","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"J.-M. Deshouillers, H. Iwaniec,An additive divisor problem, J. London Math. Soc. (2)26(1982), no. 1, 1–14","work_id":"e97054f8-51a7-4114-b637-192618e6f027","year":1982}],"snapshot_sha256":"28d60d87379a8ca4bf8b02b857178c04d51726b890c9b38010283197c4b7c8be"},"source":{"id":"2605.15434","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T14:50:10.215864Z","id":"08e2bdf6-f0da-4c2d-9e31-0e01c5381e8a","model_set":{"reader":"grok-4.3"},"one_line_summary":"Proves asymptotic count of solutions to x1 x2 - x3 x4 = h for xi in [-N, N] with square-root cancellation when h = N^2 + O(N), confirming a prior speculation.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"When h equals N squared plus O(N), the number of solutions to x1 x2 minus x3 x4 equals h inside the box of side 2N admits an asymptotic with square-root cancellation error terms.","strongest_claim":"When h = N^2 + O(N), the number of solutions admits an asymptotic formula with square-root cancellation error terms obtained by exploiting symmetry via Ramanujan sums and bypassing Kloosterman sum bounds.","weakest_assumption":"The additional symmetry present precisely when h = N^2 + O(N) permits direct use of Ramanujan sums to achieve square-root cancellation without relying on general Kloosterman bounds."}},"verdict_id":"08e2bdf6-f0da-4c2d-9e31-0e01c5381e8a"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d09673e67d342ad34e12d0d39270c4a4560113cce88dd72c0bec554fcd383b12","target":"record","created_at":"2026-05-20T00:00:58Z","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":"923c86fc0e7aea428b4c9c9477cbaeea5b08e23bd608e2c6ecfeeecc3081db03","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T21:26:51Z","title_canon_sha256":"7977e62c51996c1caf8eec07fbc57eccfc08fb03696928e4680bc6257c4017da"},"schema_version":"1.0","source":{"id":"2605.15434","kind":"arxiv","version":1}},"canonical_sha256":"d00073e744e038c5162f9c5ff89bcf6f56d591d27c31035cf61acd7ef83a2b2d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d00073e744e038c5162f9c5ff89bcf6f56d591d27c31035cf61acd7ef83a2b2d","first_computed_at":"2026-05-20T00:00:58.440008Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:00:58.440008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PozulJDRo54vzgrXz1GkysyKREYHCEi44CtP/0KIj88gwJ08EhpfOo+mLOgO/DyWAExYwkHCUOiQff2g0dgzCg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:00:58.440772Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.15434","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d09673e67d342ad34e12d0d39270c4a4560113cce88dd72c0bec554fcd383b12","sha256:c948545b55151929762d9b464aa4efd98104d42f080ebebfb7197e9b5f254368"],"state_sha256":"61e6f9993570934de3c684e55a3baa99bafbaaf78685fb5487b9c4c4be0cfcee"}