{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:XX7ELARIVGRXJ4XDDHTLJMTMRG","short_pith_number":"pith:XX7ELARI","canonical_record":{"source":{"id":"2605.27639","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-26T19:57:59Z","cross_cats_sorted":[],"title_canon_sha256":"68b404c25bb90ae3af7c84e4f73888bee2c504ffe0aa36fa09c9859df841d29e","abstract_canon_sha256":"23d233526708437ffe302ff2d15861a9ba7cdc1c372b7129a460c8c4975fa174"},"schema_version":"1.0"},"canonical_sha256":"bdfe458228a9a374f2e319e6b4b26c89b6a92fb12349c0141ae9c9c9b9c101a4","source":{"kind":"arxiv","id":"2605.27639","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.27639","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"arxiv_version","alias_value":"2605.27639v1","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.27639","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"pith_short_12","alias_value":"XX7ELARIVGRX","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"pith_short_16","alias_value":"XX7ELARIVGRXJ4XD","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"pith_short_8","alias_value":"XX7ELARI","created_at":"2026-05-28T01:04:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:XX7ELARIVGRXJ4XDDHTLJMTMRG","target":"record","payload":{"canonical_record":{"source":{"id":"2605.27639","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-26T19:57:59Z","cross_cats_sorted":[],"title_canon_sha256":"68b404c25bb90ae3af7c84e4f73888bee2c504ffe0aa36fa09c9859df841d29e","abstract_canon_sha256":"23d233526708437ffe302ff2d15861a9ba7cdc1c372b7129a460c8c4975fa174"},"schema_version":"1.0"},"canonical_sha256":"bdfe458228a9a374f2e319e6b4b26c89b6a92fb12349c0141ae9c9c9b9c101a4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T01:04:45.028177Z","signature_b64":"ARm1oz0qGqndC4YceQglgwTMNa4sWR6uCI+24eW/zRFkcCt+eI8sA20b2AKz9LhQgr4OyTwORWzaV/KUMPEMDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdfe458228a9a374f2e319e6b4b26c89b6a92fb12349c0141ae9c9c9b9c101a4","last_reissued_at":"2026-05-28T01:04:45.027711Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T01:04:45.027711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.27639","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-28T01:04:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LknnvU1JQXeaqsKnnTEHGxJHrranZFLhfhknJKWih3VJ36GzT3WFHpZOdHZNgJYhEoWsf9mAaZLIwXBnDbjTCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T12:28:34.541906Z"},"content_sha256":"42a4563ed226cbecfca3624cdaf2877856fb4c23ec23e90489a0a64351a6f7ed","schema_version":"1.0","event_id":"sha256:42a4563ed226cbecfca3624cdaf2877856fb4c23ec23e90489a0a64351a6f7ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:XX7ELARIVGRXJ4XDDHTLJMTMRG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some Remarks on $\\tau$-Congruent Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Debajyoti De, Shamik Das","submitted_at":"2026-05-26T19:57:59Z","abstract_excerpt":"In this paper, we extend the work of \\cite{Chahal} in several directions. We first determine all Heron triangles that tightly circumscribe the unit circle and the associated $\\tau$-congruent numbers generated by them. We then characterize all rational right triangles that tightly circumscribe the unit ellipse and identify the corresponding congruent numbers. In addition, we study of the congruent numbers from the excircle opposite a vertex of a rational right triangle, that is, the circle tangent to one side of the triangle and to the extensions of the remaining two sides."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.27639","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/2605.27639/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-28T01:04:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ypwHmUKJ6ne/ei0GtWsBI3ILRQiJvq9+RiwNlQXBoZxyMQukWlIjPdLZuLNes4ME2HdQgS2lAo3871drKsRTBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T12:28:34.542292Z"},"content_sha256":"03d92375e057c58de28822563aaf20099c4c6576e3d933a3b73d0d8c24a8c536","schema_version":"1.0","event_id":"sha256:03d92375e057c58de28822563aaf20099c4c6576e3d933a3b73d0d8c24a8c536"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XX7ELARIVGRXJ4XDDHTLJMTMRG/bundle.json","state_url":"https://pith.science/pith/XX7ELARIVGRXJ4XDDHTLJMTMRG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XX7ELARIVGRXJ4XDDHTLJMTMRG/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":"2026-06-26T12:28:34Z","links":{"resolver":"https://pith.science/pith/XX7ELARIVGRXJ4XDDHTLJMTMRG","bundle":"https://pith.science/pith/XX7ELARIVGRXJ4XDDHTLJMTMRG/bundle.json","state":"https://pith.science/pith/XX7ELARIVGRXJ4XDDHTLJMTMRG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XX7ELARIVGRXJ4XDDHTLJMTMRG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:XX7ELARIVGRXJ4XDDHTLJMTMRG","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":"23d233526708437ffe302ff2d15861a9ba7cdc1c372b7129a460c8c4975fa174","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-26T19:57:59Z","title_canon_sha256":"68b404c25bb90ae3af7c84e4f73888bee2c504ffe0aa36fa09c9859df841d29e"},"schema_version":"1.0","source":{"id":"2605.27639","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.27639","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"arxiv_version","alias_value":"2605.27639v1","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.27639","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"pith_short_12","alias_value":"XX7ELARIVGRX","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"pith_short_16","alias_value":"XX7ELARIVGRXJ4XD","created_at":"2026-05-28T01:04:45Z"},{"alias_kind":"pith_short_8","alias_value":"XX7ELARI","created_at":"2026-05-28T01:04:45Z"}],"graph_snapshots":[{"event_id":"sha256:03d92375e057c58de28822563aaf20099c4c6576e3d933a3b73d0d8c24a8c536","target":"graph","created_at":"2026-05-28T01:04:45Z","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/2605.27639/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we extend the work of \\cite{Chahal} in several directions. We first determine all Heron triangles that tightly circumscribe the unit circle and the associated $\\tau$-congruent numbers generated by them. We then characterize all rational right triangles that tightly circumscribe the unit ellipse and identify the corresponding congruent numbers. In addition, we study of the congruent numbers from the excircle opposite a vertex of a rational right triangle, that is, the circle tangent to one side of the triangle and to the extensions of the remaining two sides.","authors_text":"Debajyoti De, Shamik Das","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-26T19:57:59Z","title":"Some Remarks on $\\tau$-Congruent Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.27639","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":"sha256:42a4563ed226cbecfca3624cdaf2877856fb4c23ec23e90489a0a64351a6f7ed","target":"record","created_at":"2026-05-28T01:04:45Z","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":"23d233526708437ffe302ff2d15861a9ba7cdc1c372b7129a460c8c4975fa174","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-26T19:57:59Z","title_canon_sha256":"68b404c25bb90ae3af7c84e4f73888bee2c504ffe0aa36fa09c9859df841d29e"},"schema_version":"1.0","source":{"id":"2605.27639","kind":"arxiv","version":1}},"canonical_sha256":"bdfe458228a9a374f2e319e6b4b26c89b6a92fb12349c0141ae9c9c9b9c101a4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdfe458228a9a374f2e319e6b4b26c89b6a92fb12349c0141ae9c9c9b9c101a4","first_computed_at":"2026-05-28T01:04:45.027711Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T01:04:45.027711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ARm1oz0qGqndC4YceQglgwTMNa4sWR6uCI+24eW/zRFkcCt+eI8sA20b2AKz9LhQgr4OyTwORWzaV/KUMPEMDg==","signature_status":"signed_v1","signed_at":"2026-05-28T01:04:45.028177Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.27639","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42a4563ed226cbecfca3624cdaf2877856fb4c23ec23e90489a0a64351a6f7ed","sha256:03d92375e057c58de28822563aaf20099c4c6576e3d933a3b73d0d8c24a8c536"],"state_sha256":"c123868df3e07e6828cdfae754c3b839e2ea96a2301fa0fad775a4913e0af66f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8eevubCAeWYiBIg9Row0040+nWg0vlBbP1RzqZnuwWz+YOrFzFU3khzyWZTCrI6tUdoyradjpUMktUW+NcpOAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T12:28:34.544456Z","bundle_sha256":"02104254c1719c74c086eabed555add0c17c819e235f8a517e5a7b19d38fb2a0"}}