{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:5GM5H3DZD7UAVTOFJZHGCRJONY","short_pith_number":"pith:5GM5H3DZ","canonical_record":{"source":{"id":"0903.2518","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-03-14T00:31:05Z","cross_cats_sorted":[],"title_canon_sha256":"f66c1ca6120f921116f3f7fc30fe452f9dbe9a35c57eddf566d17d71faf9adb5","abstract_canon_sha256":"717aa80a137f95f476dafe862cd566d713ba453e69739f7e6582e0b1a18ff668"},"schema_version":"1.0"},"canonical_sha256":"e999d3ec791fe80acdc54e4e61452e6e29e618fec8888bf7273f109707208733","source":{"kind":"arxiv","id":"0903.2518","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.2518","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"arxiv_version","alias_value":"0903.2518v3","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.2518","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"pith_short_12","alias_value":"5GM5H3DZD7UA","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"pith_short_16","alias_value":"5GM5H3DZD7UAVTOF","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"pith_short_8","alias_value":"5GM5H3DZ","created_at":"2026-07-04T17:27:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:5GM5H3DZD7UAVTOFJZHGCRJONY","target":"record","payload":{"canonical_record":{"source":{"id":"0903.2518","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-03-14T00:31:05Z","cross_cats_sorted":[],"title_canon_sha256":"f66c1ca6120f921116f3f7fc30fe452f9dbe9a35c57eddf566d17d71faf9adb5","abstract_canon_sha256":"717aa80a137f95f476dafe862cd566d713ba453e69739f7e6582e0b1a18ff668"},"schema_version":"1.0"},"canonical_sha256":"e999d3ec791fe80acdc54e4e61452e6e29e618fec8888bf7273f109707208733","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T17:27:53.125293Z","signature_b64":"2EdSwQn8j8qoXuHKbdlZ/YO9oM651MOiksMax706FMBm7xzTTr9RrOWABDkrg9zauKx2sCQ3orh0MCfJ8ASqBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e999d3ec791fe80acdc54e4e61452e6e29e618fec8888bf7273f109707208733","last_reissued_at":"2026-07-04T17:27:53.124890Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T17:27:53.124890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.2518","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T17:27:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Cr5Tl3qUEXNB/pzBnlMdT1aKHOn09Z1uiD5+EkPGuVppqxxs7AvVIS/JVbiIZgst/0hgwPPr1YhX8zFRwEJmCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T16:25:25.204949Z"},"content_sha256":"bc0fbfcf7101c0773b5e45a5483755c0fb5ee3adebe9b00e0e07cce0ef038624","schema_version":"1.0","event_id":"sha256:bc0fbfcf7101c0773b5e45a5483755c0fb5ee3adebe9b00e0e07cce0ef038624"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:5GM5H3DZD7UAVTOFJZHGCRJONY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A remainder estimate for Weyl's law on Liouville tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Hugues Lapointe","submitted_at":"2009-03-14T00:31:05Z","abstract_excerpt":"The paper is concerned with the asymptotic distribution of Laplace eigenvalues on Liouville tori. Liouville metrics are the largest known class of integrable metrics on two-dimensional tori; they contain flat metrics and metrics of revolution as special cases. Using separation of variables, we reduce the eigenvalue counting problem to the problem of counting lattice points in certain planar domains. This allows us to improve the remainder estimate in Weyl's law on a large class of Liouville tori. For flat metrics, such an estimate has been known for more than a century due to classical results"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.2518","kind":"arxiv","version":3},"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/0903.2518/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":"2026-07-04T17:27:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xW38qD6AVo0ljmRaSEaaCWXxnz09E1ETDrwIaDICiNQAesGQY1s/5COaEfRP1QHtWrZQBSpW8PqU2zYUQ4eMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T16:25:25.205613Z"},"content_sha256":"268e8d9df2f804ebeecfffdb069c2c1dd199cd63c1df55466ca90fa65f38b50e","schema_version":"1.0","event_id":"sha256:268e8d9df2f804ebeecfffdb069c2c1dd199cd63c1df55466ca90fa65f38b50e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5GM5H3DZD7UAVTOFJZHGCRJONY/bundle.json","state_url":"https://pith.science/pith/5GM5H3DZD7UAVTOFJZHGCRJONY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5GM5H3DZD7UAVTOFJZHGCRJONY/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":"2026-08-10T16:25:25Z","links":{"resolver":"https://pith.science/pith/5GM5H3DZD7UAVTOFJZHGCRJONY","bundle":"https://pith.science/pith/5GM5H3DZD7UAVTOFJZHGCRJONY/bundle.json","state":"https://pith.science/pith/5GM5H3DZD7UAVTOFJZHGCRJONY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5GM5H3DZD7UAVTOFJZHGCRJONY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:5GM5H3DZD7UAVTOFJZHGCRJONY","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":"717aa80a137f95f476dafe862cd566d713ba453e69739f7e6582e0b1a18ff668","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-03-14T00:31:05Z","title_canon_sha256":"f66c1ca6120f921116f3f7fc30fe452f9dbe9a35c57eddf566d17d71faf9adb5"},"schema_version":"1.0","source":{"id":"0903.2518","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.2518","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"arxiv_version","alias_value":"0903.2518v3","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.2518","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"pith_short_12","alias_value":"5GM5H3DZD7UA","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"pith_short_16","alias_value":"5GM5H3DZD7UAVTOF","created_at":"2026-07-04T17:27:53Z"},{"alias_kind":"pith_short_8","alias_value":"5GM5H3DZ","created_at":"2026-07-04T17:27:53Z"}],"graph_snapshots":[{"event_id":"sha256:268e8d9df2f804ebeecfffdb069c2c1dd199cd63c1df55466ca90fa65f38b50e","target":"graph","created_at":"2026-07-04T17:27:53Z","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/0903.2518/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The paper is concerned with the asymptotic distribution of Laplace eigenvalues on Liouville tori. Liouville metrics are the largest known class of integrable metrics on two-dimensional tori; they contain flat metrics and metrics of revolution as special cases. Using separation of variables, we reduce the eigenvalue counting problem to the problem of counting lattice points in certain planar domains. This allows us to improve the remainder estimate in Weyl's law on a large class of Liouville tori. For flat metrics, such an estimate has been known for more than a century due to classical results","authors_text":"Hugues Lapointe","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-03-14T00:31:05Z","title":"A remainder estimate for Weyl's law on Liouville tori"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.2518","kind":"arxiv","version":3},"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:bc0fbfcf7101c0773b5e45a5483755c0fb5ee3adebe9b00e0e07cce0ef038624","target":"record","created_at":"2026-07-04T17:27:53Z","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":"717aa80a137f95f476dafe862cd566d713ba453e69739f7e6582e0b1a18ff668","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-03-14T00:31:05Z","title_canon_sha256":"f66c1ca6120f921116f3f7fc30fe452f9dbe9a35c57eddf566d17d71faf9adb5"},"schema_version":"1.0","source":{"id":"0903.2518","kind":"arxiv","version":3}},"canonical_sha256":"e999d3ec791fe80acdc54e4e61452e6e29e618fec8888bf7273f109707208733","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e999d3ec791fe80acdc54e4e61452e6e29e618fec8888bf7273f109707208733","first_computed_at":"2026-07-04T17:27:53.124890Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T17:27:53.124890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2EdSwQn8j8qoXuHKbdlZ/YO9oM651MOiksMax706FMBm7xzTTr9RrOWABDkrg9zauKx2sCQ3orh0MCfJ8ASqBQ==","signature_status":"signed_v1","signed_at":"2026-07-04T17:27:53.125293Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.2518","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bc0fbfcf7101c0773b5e45a5483755c0fb5ee3adebe9b00e0e07cce0ef038624","sha256:268e8d9df2f804ebeecfffdb069c2c1dd199cd63c1df55466ca90fa65f38b50e"],"state_sha256":"65c485a884c87e5d4c607f0068d04cdafee2783731525a482a20c1320908d4c7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MTEOKOz/e6FGIhoduJWpkaSy1NjCaXFz0JMstypIFgTfFJzkVJmJMPmg7kOBYmx3eD9pZHAdTPxsS3MwpwZyBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T16:25:25.210821Z","bundle_sha256":"4cc2a0aded8ca1f886bb48174ed8429f60f178d9546f09184d0162afc0391ca7"}}