{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:2KUFMYEIQU6GMPQ47JKE2X3CWW","short_pith_number":"pith:2KUFMYEI","canonical_record":{"source":{"id":"math/0412409","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2004-12-20T18:15:16Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"75f37ef8159e526a9031474e1ba660679204d7a0af415d501a8a6eed27767404","abstract_canon_sha256":"ef8ed75c08568e8d3c5163f1d8bf20d4364528d6d27a24113f03afcc8f7489d0"},"schema_version":"1.0"},"canonical_sha256":"d2a8566088853c663e1cfa544d5f62b5a029df77e934c26ffc9c6370878f581b","source":{"kind":"arxiv","id":"math/0412409","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0412409","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"arxiv_version","alias_value":"math/0412409v2","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0412409","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"pith_short_12","alias_value":"2KUFMYEIQU6G","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"2KUFMYEIQU6GMPQ4","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"2KUFMYEI","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:2KUFMYEIQU6GMPQ47JKE2X3CWW","target":"record","payload":{"canonical_record":{"source":{"id":"math/0412409","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2004-12-20T18:15:16Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"75f37ef8159e526a9031474e1ba660679204d7a0af415d501a8a6eed27767404","abstract_canon_sha256":"ef8ed75c08568e8d3c5163f1d8bf20d4364528d6d27a24113f03afcc8f7489d0"},"schema_version":"1.0"},"canonical_sha256":"d2a8566088853c663e1cfa544d5f62b5a029df77e934c26ffc9c6370878f581b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:27.000789Z","signature_b64":"8kGrcg1nVUgXsjNOUE//ZtbUpD9zwuFsvd1fdbfKNQzt2cKqAwrKBnvZGOidOvzWu3lIB8Pt2m/kl3NfbhGeCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2a8566088853c663e1cfa544d5f62b5a029df77e934c26ffc9c6370878f581b","last_reissued_at":"2026-05-18T01:38:27.000226Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:27.000226Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0412409","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:38:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lkogVlvkpw6Yx3FAqxYq/GstK9L4Zukw20DcIDfCiMuAcLACyUMLHW9QLa1UlyT2N/KhAAsCIY/VcD4krHXaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T18:37:54.690670Z"},"content_sha256":"faf62b60d021e721cd6da839a89829df363cac17557b474f1074543bb016013c","schema_version":"1.0","event_id":"sha256:faf62b60d021e721cd6da839a89829df363cac17557b474f1074543bb016013c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:2KUFMYEIQU6GMPQ47JKE2X3CWW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The first conformal Dirac eigenvalue on 2-dimensional tori","license":"","headline":"","cross_cats":["math.SP"],"primary_cat":"math.DG","authors_text":"Bernd Ammann, Emmanuel Humbert","submitted_at":"2004-12-20T18:15:16Z","abstract_excerpt":"Let M be a compact manifold with a spin structure \\chi and a Riemannian metric g. Let \\lambda_g^2 be the smallest eigenvalue of the square of the Dirac operator with respect to g and \\chi. The \\tau-invariant is defined as \\tau(M,\\chi):= sup inf \\sqrt{\\lambda_g^2} Vol(M,g)^{1/n} where the supremum runs over the set of all conformal classes on M, and where the infimum runs over all metrics in the given class. We show that \\tau(T^2,\\chi)=2\\sqrt{\\pi} if \\chi is ``the'' non-trivial spin structure on T^2. In order to calculate this invariant, we study the infimum as a function on the spin-conformal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0412409","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T01:38:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uIJh3vrxr7B6WGp45D4ItqprsWYHzW7jFVlCFbIi8/GnDphvS+NLLPqFFKYtnDeZKHkkVOZ1TSIzpCRkv8mCAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T18:37:54.691378Z"},"content_sha256":"39f1f6da43688945efd8332592e3b83cc4b36334a8518b4cba59202ad5233fa0","schema_version":"1.0","event_id":"sha256:39f1f6da43688945efd8332592e3b83cc4b36334a8518b4cba59202ad5233fa0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2KUFMYEIQU6GMPQ47JKE2X3CWW/bundle.json","state_url":"https://pith.science/pith/2KUFMYEIQU6GMPQ47JKE2X3CWW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2KUFMYEIQU6GMPQ47JKE2X3CWW/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":"2026-05-27T18:37:54Z","links":{"resolver":"https://pith.science/pith/2KUFMYEIQU6GMPQ47JKE2X3CWW","bundle":"https://pith.science/pith/2KUFMYEIQU6GMPQ47JKE2X3CWW/bundle.json","state":"https://pith.science/pith/2KUFMYEIQU6GMPQ47JKE2X3CWW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2KUFMYEIQU6GMPQ47JKE2X3CWW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:2KUFMYEIQU6GMPQ47JKE2X3CWW","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":"ef8ed75c08568e8d3c5163f1d8bf20d4364528d6d27a24113f03afcc8f7489d0","cross_cats_sorted":["math.SP"],"license":"","primary_cat":"math.DG","submitted_at":"2004-12-20T18:15:16Z","title_canon_sha256":"75f37ef8159e526a9031474e1ba660679204d7a0af415d501a8a6eed27767404"},"schema_version":"1.0","source":{"id":"math/0412409","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0412409","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"arxiv_version","alias_value":"math/0412409v2","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0412409","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"pith_short_12","alias_value":"2KUFMYEIQU6G","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"2KUFMYEIQU6GMPQ4","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"2KUFMYEI","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:39f1f6da43688945efd8332592e3b83cc4b36334a8518b4cba59202ad5233fa0","target":"graph","created_at":"2026-05-18T01:38:27Z","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"},"paper":{"abstract_excerpt":"Let M be a compact manifold with a spin structure \\chi and a Riemannian metric g. Let \\lambda_g^2 be the smallest eigenvalue of the square of the Dirac operator with respect to g and \\chi. The \\tau-invariant is defined as \\tau(M,\\chi):= sup inf \\sqrt{\\lambda_g^2} Vol(M,g)^{1/n} where the supremum runs over the set of all conformal classes on M, and where the infimum runs over all metrics in the given class. We show that \\tau(T^2,\\chi)=2\\sqrt{\\pi} if \\chi is ``the'' non-trivial spin structure on T^2. In order to calculate this invariant, we study the infimum as a function on the spin-conformal ","authors_text":"Bernd Ammann, Emmanuel Humbert","cross_cats":["math.SP"],"headline":"","license":"","primary_cat":"math.DG","submitted_at":"2004-12-20T18:15:16Z","title":"The first conformal Dirac eigenvalue on 2-dimensional tori"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0412409","kind":"arxiv","version":2},"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:faf62b60d021e721cd6da839a89829df363cac17557b474f1074543bb016013c","target":"record","created_at":"2026-05-18T01:38:27Z","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":"ef8ed75c08568e8d3c5163f1d8bf20d4364528d6d27a24113f03afcc8f7489d0","cross_cats_sorted":["math.SP"],"license":"","primary_cat":"math.DG","submitted_at":"2004-12-20T18:15:16Z","title_canon_sha256":"75f37ef8159e526a9031474e1ba660679204d7a0af415d501a8a6eed27767404"},"schema_version":"1.0","source":{"id":"math/0412409","kind":"arxiv","version":2}},"canonical_sha256":"d2a8566088853c663e1cfa544d5f62b5a029df77e934c26ffc9c6370878f581b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d2a8566088853c663e1cfa544d5f62b5a029df77e934c26ffc9c6370878f581b","first_computed_at":"2026-05-18T01:38:27.000226Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:27.000226Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8kGrcg1nVUgXsjNOUE//ZtbUpD9zwuFsvd1fdbfKNQzt2cKqAwrKBnvZGOidOvzWu3lIB8Pt2m/kl3NfbhGeCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:27.000789Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0412409","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:faf62b60d021e721cd6da839a89829df363cac17557b474f1074543bb016013c","sha256:39f1f6da43688945efd8332592e3b83cc4b36334a8518b4cba59202ad5233fa0"],"state_sha256":"c3af80950f21e132aa0759d177b3be0b31509677f297e7b8c09f1a07e0458062"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NN8rZoc16n3U9aHOmWAUldOlj2bYawW/nkRDHrw7NCB4GJw0nn7TfrkTQIHcjjMqnpS4L18J7Qk4xpw96lDECA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T18:37:54.696198Z","bundle_sha256":"f95a13804d5d0061f735c3d15f015c25743af1e80356e96a5e54cfd83e6038a0"}}