{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:KGIEK46AYNNCJCP4TYXP5Q56IT","short_pith_number":"pith:KGIEK46A","canonical_record":{"source":{"id":"1312.0038","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2013-11-29T23:45:42Z","cross_cats_sorted":[],"title_canon_sha256":"f442913b7584d92d3540261dda6d221cb6110b50bbd8da4fcbd4e9803b02886e","abstract_canon_sha256":"414b5333e701e3053f16c46c4e5c0dbdd90732c45ff04f318f47406f8cc91ce7"},"schema_version":"1.0"},"canonical_sha256":"51904573c0c35a2489fc9e2efec3be44ed41c3ef1215df6069aa4a0a8df4c19a","source":{"kind":"arxiv","id":"1312.0038","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.0038","created_at":"2026-05-18T01:46:04Z"},{"alias_kind":"arxiv_version","alias_value":"1312.0038v2","created_at":"2026-05-18T01:46:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.0038","created_at":"2026-05-18T01:46:04Z"},{"alias_kind":"pith_short_12","alias_value":"KGIEK46AYNNC","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KGIEK46AYNNCJCP4","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KGIEK46A","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:KGIEK46AYNNCJCP4TYXP5Q56IT","target":"record","payload":{"canonical_record":{"source":{"id":"1312.0038","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2013-11-29T23:45:42Z","cross_cats_sorted":[],"title_canon_sha256":"f442913b7584d92d3540261dda6d221cb6110b50bbd8da4fcbd4e9803b02886e","abstract_canon_sha256":"414b5333e701e3053f16c46c4e5c0dbdd90732c45ff04f318f47406f8cc91ce7"},"schema_version":"1.0"},"canonical_sha256":"51904573c0c35a2489fc9e2efec3be44ed41c3ef1215df6069aa4a0a8df4c19a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:46:04.642962Z","signature_b64":"LZNLtZGph4HWpfbT5ieRf5lbfCRb/CxZ+k01XuhPXnxPIxZwzHkLmkoHBQO2awJ6DCtm2WCac3vkQlusILIBCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51904573c0c35a2489fc9e2efec3be44ed41c3ef1215df6069aa4a0a8df4c19a","last_reissued_at":"2026-05-18T01:46:04.642559Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:46:04.642559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.0038","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":"2026-05-18T01:46:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v8axys8KyI7VtRn2l8mNkyJhjwcRO47yUrQD3KpEHTCJ2OR0RHasHTeQG2fGI1UnDjkDsRg92Tn0ywt9TEoAAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T08:05:39.091086Z"},"content_sha256":"effef2bd7db366973ac9f782a662b830f4f711d58f283a185364e1cdf30183be","schema_version":"1.0","event_id":"sha256:effef2bd7db366973ac9f782a662b830f4f711d58f283a185364e1cdf30183be"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:KGIEK46AYNNCJCP4TYXP5Q56IT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds on Operator Dimensions in 2D Conformal Field Theories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"Alfred D. Shapere, Joshua D. Qualls","submitted_at":"2013-11-29T23:45:42Z","abstract_excerpt":"We extend the work of Hellerman (arxiv:0902.2790) to derive an upper bound on the conformal dimension $\\Delta_2$ of the next-to-lowest nontrival primary operator in unitary two-dimensional conformal field theories without chiral primary operators. The bound we find is of the same form as found for $\\Delta_1$: $\\Delta_2 \\leq c_{tot}/12 + O(1)$. We find a similar bound on the conformal dimension $\\Delta_3$, and present a method for deriving bounds on $\\Delta_n$ for any $n$, under slightly modified assumptions. For asymptotically large $c_{tot}$ and fixed $n$, we show that $\\Delta_n \\leq \\frac{c_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0038","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":"2026-05-18T01:46:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Am1SLJEE8HEnMJiGX0SZwJWecwQflgVHUDoWSKvWZxvbsudQ0YV4WBE9BE8f3h3wAEZ3eA0qj9q4VTiuh6TsDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T08:05:39.091438Z"},"content_sha256":"8d2c03a4943ca6405a58f94d1a386f7457acc76696419093a1a1146c3c189f37","schema_version":"1.0","event_id":"sha256:8d2c03a4943ca6405a58f94d1a386f7457acc76696419093a1a1146c3c189f37"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KGIEK46AYNNCJCP4TYXP5Q56IT/bundle.json","state_url":"https://pith.science/pith/KGIEK46AYNNCJCP4TYXP5Q56IT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KGIEK46AYNNCJCP4TYXP5Q56IT/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":"2026-06-28T08:05:39Z","links":{"resolver":"https://pith.science/pith/KGIEK46AYNNCJCP4TYXP5Q56IT","bundle":"https://pith.science/pith/KGIEK46AYNNCJCP4TYXP5Q56IT/bundle.json","state":"https://pith.science/pith/KGIEK46AYNNCJCP4TYXP5Q56IT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KGIEK46AYNNCJCP4TYXP5Q56IT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KGIEK46AYNNCJCP4TYXP5Q56IT","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":"414b5333e701e3053f16c46c4e5c0dbdd90732c45ff04f318f47406f8cc91ce7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2013-11-29T23:45:42Z","title_canon_sha256":"f442913b7584d92d3540261dda6d221cb6110b50bbd8da4fcbd4e9803b02886e"},"schema_version":"1.0","source":{"id":"1312.0038","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.0038","created_at":"2026-05-18T01:46:04Z"},{"alias_kind":"arxiv_version","alias_value":"1312.0038v2","created_at":"2026-05-18T01:46:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.0038","created_at":"2026-05-18T01:46:04Z"},{"alias_kind":"pith_short_12","alias_value":"KGIEK46AYNNC","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KGIEK46AYNNCJCP4","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KGIEK46A","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:8d2c03a4943ca6405a58f94d1a386f7457acc76696419093a1a1146c3c189f37","target":"graph","created_at":"2026-05-18T01:46:04Z","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":"We extend the work of Hellerman (arxiv:0902.2790) to derive an upper bound on the conformal dimension $\\Delta_2$ of the next-to-lowest nontrival primary operator in unitary two-dimensional conformal field theories without chiral primary operators. The bound we find is of the same form as found for $\\Delta_1$: $\\Delta_2 \\leq c_{tot}/12 + O(1)$. We find a similar bound on the conformal dimension $\\Delta_3$, and present a method for deriving bounds on $\\Delta_n$ for any $n$, under slightly modified assumptions. For asymptotically large $c_{tot}$ and fixed $n$, we show that $\\Delta_n \\leq \\frac{c_","authors_text":"Alfred D. Shapere, Joshua D. Qualls","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2013-11-29T23:45:42Z","title":"Bounds on Operator Dimensions in 2D Conformal Field Theories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0038","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:effef2bd7db366973ac9f782a662b830f4f711d58f283a185364e1cdf30183be","target":"record","created_at":"2026-05-18T01:46:04Z","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":"414b5333e701e3053f16c46c4e5c0dbdd90732c45ff04f318f47406f8cc91ce7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2013-11-29T23:45:42Z","title_canon_sha256":"f442913b7584d92d3540261dda6d221cb6110b50bbd8da4fcbd4e9803b02886e"},"schema_version":"1.0","source":{"id":"1312.0038","kind":"arxiv","version":2}},"canonical_sha256":"51904573c0c35a2489fc9e2efec3be44ed41c3ef1215df6069aa4a0a8df4c19a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"51904573c0c35a2489fc9e2efec3be44ed41c3ef1215df6069aa4a0a8df4c19a","first_computed_at":"2026-05-18T01:46:04.642559Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:46:04.642559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LZNLtZGph4HWpfbT5ieRf5lbfCRb/CxZ+k01XuhPXnxPIxZwzHkLmkoHBQO2awJ6DCtm2WCac3vkQlusILIBCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:46:04.642962Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.0038","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:effef2bd7db366973ac9f782a662b830f4f711d58f283a185364e1cdf30183be","sha256:8d2c03a4943ca6405a58f94d1a386f7457acc76696419093a1a1146c3c189f37"],"state_sha256":"f32e7f1bd51adae2328322e27499c5ccb8759b516707d1b29f498ba8a2285b7c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O/GcVW3naXZTYY4IMK5bvLzVz/jGLBnqtpB2ejEn7Xb5zKqrBdtLTMMIoWVFZ+eZMNyZWIgmdJ6DiWeXei+9Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T08:05:39.093404Z","bundle_sha256":"7ae175bfa518947c27875d4d1cd958fb7c45bff54ff3235433998ccdfaa8047c"}}