{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:A4OEHJQ7TWHY2K6SQRJJX4R4R4","short_pith_number":"pith:A4OEHJQ7","canonical_record":{"source":{"id":"2411.19864","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.HO","submitted_at":"2024-11-29T17:27:00Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e598e4df4eadb3c9387387bddc354e04cf72bf772316506197afac0f1a08fb51","abstract_canon_sha256":"cba3f1067e134180d4f80862d487027d4796706434b5fb7a546f013571c9d750"},"schema_version":"1.0"},"canonical_sha256":"071c43a61f9d8f8d2bd284529bf23c8f3d4c47355d347d51be508dcc7e6883f4","source":{"kind":"arxiv","id":"2411.19864","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.19864","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"arxiv_version","alias_value":"2411.19864v4","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.19864","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"pith_short_12","alias_value":"A4OEHJQ7TWHY","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"pith_short_16","alias_value":"A4OEHJQ7TWHY2K6S","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"pith_short_8","alias_value":"A4OEHJQ7","created_at":"2026-06-03T01:05:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:A4OEHJQ7TWHY2K6SQRJJX4R4R4","target":"record","payload":{"canonical_record":{"source":{"id":"2411.19864","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.HO","submitted_at":"2024-11-29T17:27:00Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e598e4df4eadb3c9387387bddc354e04cf72bf772316506197afac0f1a08fb51","abstract_canon_sha256":"cba3f1067e134180d4f80862d487027d4796706434b5fb7a546f013571c9d750"},"schema_version":"1.0"},"canonical_sha256":"071c43a61f9d8f8d2bd284529bf23c8f3d4c47355d347d51be508dcc7e6883f4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:03.777883Z","signature_b64":"7J+bzPgwlMtRKSpE9UK0Hw4AgevTzbECtJ7MKHPU3xosRgqBn7c4SkNijOn2XVfScskrdumWNA1Oa75y6mNiAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"071c43a61f9d8f8d2bd284529bf23c8f3d4c47355d347d51be508dcc7e6883f4","last_reissued_at":"2026-06-03T01:05:03.777353Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:03.777353Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.19864","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":"2026-06-03T01:05:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eF2EJ+ROqeYwWKKwqF9V0ol5pn+cMOQd2zbDeIJtkEkW0FsZOR71pWjdmSvoafHzwY0Q6CR2dW8a2ScYv+9oAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T02:00:25.098208Z"},"content_sha256":"c0e9af4517d01a19c4ac31999430c48209cc04d7f725f7bc9967a83979b74e9a","schema_version":"1.0","event_id":"sha256:c0e9af4517d01a19c4ac31999430c48209cc04d7f725f7bc9967a83979b74e9a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:A4OEHJQ7TWHY2K6SQRJJX4R4R4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.HO","authors_text":"Muthu Veerappan Ramalingam, Zbigniew Fiedorowicz","submitted_at":"2024-11-29T17:27:00Z","abstract_excerpt":"It is well known that there is a somewhat mysterious relation between the area of the quartic Fermat curve $x^4+y^4=1$, aka squircle, and the arc length of the lemniscate $(x^2+y^2)^2=x^2-y^2$. The standardproof of this fact uses relations between elliptic integrals and the gamma function. In this article we generalize this result to relate areas of sectors of the squircle to arc lengths of segments of the lemniscate. We provide a geometric interpretation of this relation and an elementary proof of the relation, which only uses basic integral calculus. We also discuss an alternate version of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.19864","kind":"arxiv","version":4},"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/2411.19864/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":"2026-06-03T01:05:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XMQpzIdmMIqDLoghvfdoQ1It1b/clmtsNnKXj380w3DdFAfCqccVUPekXMS5NC4R/DSYxVlW4wWt1hJoRTRwDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T02:00:25.098751Z"},"content_sha256":"bab96f27b22492ab63944f177d48ce92ee0cef5c5148c02bfdb0ef62a01ab539","schema_version":"1.0","event_id":"sha256:bab96f27b22492ab63944f177d48ce92ee0cef5c5148c02bfdb0ef62a01ab539"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A4OEHJQ7TWHY2K6SQRJJX4R4R4/bundle.json","state_url":"https://pith.science/pith/A4OEHJQ7TWHY2K6SQRJJX4R4R4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A4OEHJQ7TWHY2K6SQRJJX4R4R4/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":"2026-08-19T02:00:25Z","links":{"resolver":"https://pith.science/pith/A4OEHJQ7TWHY2K6SQRJJX4R4R4","bundle":"https://pith.science/pith/A4OEHJQ7TWHY2K6SQRJJX4R4R4/bundle.json","state":"https://pith.science/pith/A4OEHJQ7TWHY2K6SQRJJX4R4R4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A4OEHJQ7TWHY2K6SQRJJX4R4R4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:A4OEHJQ7TWHY2K6SQRJJX4R4R4","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":"cba3f1067e134180d4f80862d487027d4796706434b5fb7a546f013571c9d750","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.HO","submitted_at":"2024-11-29T17:27:00Z","title_canon_sha256":"e598e4df4eadb3c9387387bddc354e04cf72bf772316506197afac0f1a08fb51"},"schema_version":"1.0","source":{"id":"2411.19864","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.19864","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"arxiv_version","alias_value":"2411.19864v4","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.19864","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"pith_short_12","alias_value":"A4OEHJQ7TWHY","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"pith_short_16","alias_value":"A4OEHJQ7TWHY2K6S","created_at":"2026-06-03T01:05:03Z"},{"alias_kind":"pith_short_8","alias_value":"A4OEHJQ7","created_at":"2026-06-03T01:05:03Z"}],"graph_snapshots":[{"event_id":"sha256:bab96f27b22492ab63944f177d48ce92ee0cef5c5148c02bfdb0ef62a01ab539","target":"graph","created_at":"2026-06-03T01:05:03Z","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/2411.19864/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"It is well known that there is a somewhat mysterious relation between the area of the quartic Fermat curve $x^4+y^4=1$, aka squircle, and the arc length of the lemniscate $(x^2+y^2)^2=x^2-y^2$. The standardproof of this fact uses relations between elliptic integrals and the gamma function. In this article we generalize this result to relate areas of sectors of the squircle to arc lengths of segments of the lemniscate. We provide a geometric interpretation of this relation and an elementary proof of the relation, which only uses basic integral calculus. We also discuss an alternate version of t","authors_text":"Muthu Veerappan Ramalingam, Zbigniew Fiedorowicz","cross_cats":["math.NT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.HO","submitted_at":"2024-11-29T17:27:00Z","title":"An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.19864","kind":"arxiv","version":4},"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:c0e9af4517d01a19c4ac31999430c48209cc04d7f725f7bc9967a83979b74e9a","target":"record","created_at":"2026-06-03T01:05:03Z","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":"cba3f1067e134180d4f80862d487027d4796706434b5fb7a546f013571c9d750","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.HO","submitted_at":"2024-11-29T17:27:00Z","title_canon_sha256":"e598e4df4eadb3c9387387bddc354e04cf72bf772316506197afac0f1a08fb51"},"schema_version":"1.0","source":{"id":"2411.19864","kind":"arxiv","version":4}},"canonical_sha256":"071c43a61f9d8f8d2bd284529bf23c8f3d4c47355d347d51be508dcc7e6883f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"071c43a61f9d8f8d2bd284529bf23c8f3d4c47355d347d51be508dcc7e6883f4","first_computed_at":"2026-06-03T01:05:03.777353Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T01:05:03.777353Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7J+bzPgwlMtRKSpE9UK0Hw4AgevTzbECtJ7MKHPU3xosRgqBn7c4SkNijOn2XVfScskrdumWNA1Oa75y6mNiAw==","signature_status":"signed_v1","signed_at":"2026-06-03T01:05:03.777883Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.19864","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c0e9af4517d01a19c4ac31999430c48209cc04d7f725f7bc9967a83979b74e9a","sha256:bab96f27b22492ab63944f177d48ce92ee0cef5c5148c02bfdb0ef62a01ab539"],"state_sha256":"68c1b973ae527d23d2ed3f47358db2ed012c3a18d872d7243b2c6b6bdf0f0983"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SPbOvUGTobJKgocMdwO8KoZQxvAiwr+z1fPRyGQNO/9E7m3fzHDtZ6XjxhjkzKYGx4ziKkjVnqCecn/ec0pSAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T02:00:25.103381Z","bundle_sha256":"1b1f91f033d9037fb88161e0b3dc9f04fd90018a57a4f77a0b63e8f1b959f972"}}