{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5LTQ6BFVVUSM2SOYQ4EXWBIGWX","short_pith_number":"pith:5LTQ6BFV","canonical_record":{"source":{"id":"2604.20653","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-22T15:03:43Z","cross_cats_sorted":[],"title_canon_sha256":"8ea1ed3ee870ab73d575a2707fb5f52d8bead08c41e884785017d92935fa5857","abstract_canon_sha256":"769109c5a53bd119e4ed9dc12e7aee3ee38aa006a52edbb742891e12efc70ca6"},"schema_version":"1.0"},"canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","source":{"kind":"arxiv","id":"2604.20653","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.20653","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"arxiv_version","alias_value":"2604.20653v4","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.20653","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"pith_short_12","alias_value":"5LTQ6BFVVUSM","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"pith_short_16","alias_value":"5LTQ6BFVVUSM2SOY","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"pith_short_8","alias_value":"5LTQ6BFV","created_at":"2026-05-26T02:04:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5LTQ6BFVVUSM2SOYQ4EXWBIGWX","target":"record","payload":{"canonical_record":{"source":{"id":"2604.20653","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-22T15:03:43Z","cross_cats_sorted":[],"title_canon_sha256":"8ea1ed3ee870ab73d575a2707fb5f52d8bead08c41e884785017d92935fa5857","abstract_canon_sha256":"769109c5a53bd119e4ed9dc12e7aee3ee38aa006a52edbb742891e12efc70ca6"},"schema_version":"1.0"},"canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:11.176968Z","signature_b64":"8IW3rGOsQPoOURH2+mHWIyZR6SERTQ0NbG376caLJ7KUfS1A9x6JHgrXJxcRGom5gSKNDNKXY74SBw/oCXWADg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","last_reissued_at":"2026-05-26T02:04:11.176201Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:11.176201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.20653","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-26T02:04:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DZgXf4PDq1r8mroBcTo+O/Yl9ynJmMdkVjtgNkHHc+6btyPzC/Nz7X5YKtIaanGcWwk1HZmsd2hMI9I8928JAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:02:12.484289Z"},"content_sha256":"22969c164c9576bec12c840a18a76652b3687092b5b0bcd923c3ad383191e282","schema_version":"1.0","event_id":"sha256:22969c164c9576bec12c840a18a76652b3687092b5b0bcd923c3ad383191e282"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5LTQ6BFVVUSM2SOYQ4EXWBIGWX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On sums of two squares and a basis of order $2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Artyom Radomskii","submitted_at":"2026-04-22T15:03:43Z","abstract_excerpt":"Let $\\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \\nmid b$. We show that for sufficiently large positive integer $N$ there are two strings of consecutive positive integers $I_{1}=\\{n_1-m,\\ldots, n_1+m\\}$ and $I_{2}=\\{n_2-m, \\ldots, n_2+m\\}$ such that $m = [(\\log N) (\\log \\log N)^{1/325565}]$, $I_{1}\\cup I_{2} \\subset [1, N]$, $N = n_1 + n_2$, and for any $n\\in I_{1}\\cup I_{2}$ at least one of $n$ or $an+b$ does not lie in $\\mathcal{R}$. In particular, we have $n(an+b)\\notin \\mathcal{R"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For sufficiently large positive integer N there are two strings of consecutive positive integers I1={n1-m,…,n1+m} and I2={n2-m,…,n2+m} such that m=[(log N)(log log N)^{1/325565}], I1∪I2⊂[1,N], N=n1+n2, and for any n∈I1∪I2 at least one of n or an+b does not lie in R.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence holds only for sufficiently large N; the specific tiny exponent 1/325565 is an effective constant arising from analytic estimates whose validity for all large N is assumed but not verified in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For large N there exist paired intervals I1 and I2 of length ~log N (log log N)^{1/325565} with n1 + n2 = N where no n has both n and an+b as primitive sums of two squares.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7a73aeaa8cbad0a6faac9c59c89fdadcdd68562df35d4e6802a86b9fb2077b78"},"source":{"id":"2604.20653","kind":"arxiv","version":4},"verdict":{"id":"40d3e1fd-4116-4b13-9f38-80c50c14211e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T23:05:04.301176Z","strongest_claim":"For sufficiently large positive integer N there are two strings of consecutive positive integers I1={n1-m,…,n1+m} and I2={n2-m,…,n2+m} such that m=[(log N)(log log N)^{1/325565}], I1∪I2⊂[1,N], N=n1+n2, and for any n∈I1∪I2 at least one of n or an+b does not lie in R.","one_line_summary":"For large N there exist paired intervals I1 and I2 of length ~log N (log log N)^{1/325565} with n1 + n2 = N where no n has both n and an+b as primitive sums of two squares.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence holds only for sufficiently large N; the specific tiny exponent 1/325565 is an effective constant arising from analytic estimates whose validity for all large N is assumed but not verified in the abstract.","pith_extraction_headline":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.20653/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T14:35:22.657108Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T01:41:15.484825Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"426387748644c9669d54b9eb99f3529e9543cdece8000d74882613706545b67f"},"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"},"verdict_id":"40d3e1fd-4116-4b13-9f38-80c50c14211e"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-26T02:04:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ISUaTJ87IRH9olQCioAebHrYes3y35KsBt0tLVJGDjfH0SEeHxnp6YIyxjpHEcaLPfsW9GbthJIAyn7BJAj5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:02:12.484935Z"},"content_sha256":"6737dc30b9ad79b5f7e57fc56c4b7c80d02145bb64b67eedeef71a0fee0b56d6","schema_version":"1.0","event_id":"sha256:6737dc30b9ad79b5f7e57fc56c4b7c80d02145bb64b67eedeef71a0fee0b56d6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/bundle.json","state_url":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T23:02:12Z","links":{"resolver":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX","bundle":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/bundle.json","state":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5LTQ6BFVVUSM2SOYQ4EXWBIGWX","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":"769109c5a53bd119e4ed9dc12e7aee3ee38aa006a52edbb742891e12efc70ca6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-22T15:03:43Z","title_canon_sha256":"8ea1ed3ee870ab73d575a2707fb5f52d8bead08c41e884785017d92935fa5857"},"schema_version":"1.0","source":{"id":"2604.20653","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.20653","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"arxiv_version","alias_value":"2604.20653v4","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.20653","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"pith_short_12","alias_value":"5LTQ6BFVVUSM","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"pith_short_16","alias_value":"5LTQ6BFVVUSM2SOY","created_at":"2026-05-26T02:04:11Z"},{"alias_kind":"pith_short_8","alias_value":"5LTQ6BFV","created_at":"2026-05-26T02:04:11Z"}],"graph_snapshots":[{"event_id":"sha256:6737dc30b9ad79b5f7e57fc56c4b7c80d02145bb64b67eedeef71a0fee0b56d6","target":"graph","created_at":"2026-05-26T02:04:11Z","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":"For sufficiently large positive integer N there are two strings of consecutive positive integers I1={n1-m,…,n1+m} and I2={n2-m,…,n2+m} such that m=[(log N)(log log N)^{1/325565}], I1∪I2⊂[1,N], N=n1+n2, and for any n∈I1∪I2 at least one of n or an+b does not lie in R."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The existence holds only for sufficiently large N; the specific tiny exponent 1/325565 is an effective constant arising from analytic estimates whose validity for all large N is assumed but not verified in the abstract."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"For large N there exist paired intervals I1 and I2 of length ~log N (log log N)^{1/325565} with n1 + n2 = N where no n has both n and an+b as primitive sums of two squares."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares."}],"snapshot_sha256":"7a73aeaa8cbad0a6faac9c59c89fdadcdd68562df35d4e6802a86b9fb2077b78"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-21T14:35:22.657108Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-20T01:41:15.484825Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.20653/integrity.json","findings":[],"snapshot_sha256":"426387748644c9669d54b9eb99f3529e9543cdece8000d74882613706545b67f","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \\nmid b$. We show that for sufficiently large positive integer $N$ there are two strings of consecutive positive integers $I_{1}=\\{n_1-m,\\ldots, n_1+m\\}$ and $I_{2}=\\{n_2-m, \\ldots, n_2+m\\}$ such that $m = [(\\log N) (\\log \\log N)^{1/325565}]$, $I_{1}\\cup I_{2} \\subset [1, N]$, $N = n_1 + n_2$, and for any $n\\in I_{1}\\cup I_{2}$ at least one of $n$ or $an+b$ does not lie in $\\mathcal{R}$. In particular, we have $n(an+b)\\notin \\mathcal{R","authors_text":"Artyom Radomskii","cross_cats":[],"headline":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-22T15:03:43Z","title":"On sums of two squares and a basis of order $2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.20653","kind":"arxiv","version":4},"verdict":{"created_at":"2026-05-09T23:05:04.301176Z","id":"40d3e1fd-4116-4b13-9f38-80c50c14211e","model_set":{"reader":"grok-4.3"},"one_line_summary":"For large N there exist paired intervals I1 and I2 of length ~log N (log log N)^{1/325565} with n1 + n2 = N where no n has both n and an+b as primitive sums of two squares.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.","strongest_claim":"For sufficiently large positive integer N there are two strings of consecutive positive integers I1={n1-m,…,n1+m} and I2={n2-m,…,n2+m} such that m=[(log N)(log log N)^{1/325565}], I1∪I2⊂[1,N], N=n1+n2, and for any n∈I1∪I2 at least one of n or an+b does not lie in R.","weakest_assumption":"The existence holds only for sufficiently large N; the specific tiny exponent 1/325565 is an effective constant arising from analytic estimates whose validity for all large N is assumed but not verified in the abstract."}},"verdict_id":"40d3e1fd-4116-4b13-9f38-80c50c14211e"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:22969c164c9576bec12c840a18a76652b3687092b5b0bcd923c3ad383191e282","target":"record","created_at":"2026-05-26T02:04:11Z","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":"769109c5a53bd119e4ed9dc12e7aee3ee38aa006a52edbb742891e12efc70ca6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-22T15:03:43Z","title_canon_sha256":"8ea1ed3ee870ab73d575a2707fb5f52d8bead08c41e884785017d92935fa5857"},"schema_version":"1.0","source":{"id":"2604.20653","kind":"arxiv","version":4}},"canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","first_computed_at":"2026-05-26T02:04:11.176201Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T02:04:11.176201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8IW3rGOsQPoOURH2+mHWIyZR6SERTQ0NbG376caLJ7KUfS1A9x6JHgrXJxcRGom5gSKNDNKXY74SBw/oCXWADg==","signature_status":"signed_v1","signed_at":"2026-05-26T02:04:11.176968Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.20653","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22969c164c9576bec12c840a18a76652b3687092b5b0bcd923c3ad383191e282","sha256:6737dc30b9ad79b5f7e57fc56c4b7c80d02145bb64b67eedeef71a0fee0b56d6"],"state_sha256":"056cf98d42711788838cffa185796e544b50f95294d3147bc8305dfd80cb2b9a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EU/03+BW1w87EIyHQ7HIY9wXK/q1+Eh/38o3kks/OfHuq0BncCyVIfL/vFD2STrPCUhL+TtS6tOoRGrfqC1DCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T23:02:12.488041Z","bundle_sha256":"c3614de60d302a118a3d37e355ba4b7906ceb23c7517a4c510739e4362b0cb12"}}