{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:BSJ7UVDKXDVTLBDGGC5FNX6CUG","short_pith_number":"pith:BSJ7UVDK","canonical_record":{"source":{"id":"2605.12958","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2026-05-13T03:43:20Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"3a458b1383cc54e28ce52fc3b74706e2fd862ce24f154c9f1e155beb215eac3f","abstract_canon_sha256":"fc4f6427719e750d4d64af90a40d8929c48bb13fce8588b54d710b056cd21c03"},"schema_version":"1.0"},"canonical_sha256":"0c93fa546ab8eb35846630ba56dfc2a193af8895375d49f7e727efbcdc437053","source":{"kind":"arxiv","id":"2605.12958","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12958","created_at":"2026-05-18T03:09:09Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12958v1","created_at":"2026-05-18T03:09:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12958","created_at":"2026-05-18T03:09:09Z"},{"alias_kind":"pith_short_12","alias_value":"BSJ7UVDKXDVT","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"BSJ7UVDKXDVTLBDG","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"BSJ7UVDK","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:BSJ7UVDKXDVTLBDGGC5FNX6CUG","target":"record","payload":{"canonical_record":{"source":{"id":"2605.12958","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2026-05-13T03:43:20Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"3a458b1383cc54e28ce52fc3b74706e2fd862ce24f154c9f1e155beb215eac3f","abstract_canon_sha256":"fc4f6427719e750d4d64af90a40d8929c48bb13fce8588b54d710b056cd21c03"},"schema_version":"1.0"},"canonical_sha256":"0c93fa546ab8eb35846630ba56dfc2a193af8895375d49f7e727efbcdc437053","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:09.238944Z","signature_b64":"7T01brJCsRA/rdvk4w0XnV9/pbfIxPf0Qp9UGpmPMRjQ+iZjrCUqOJwFfps8hD60EvX3qQeYMDlL1KQiK4HuBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c93fa546ab8eb35846630ba56dfc2a193af8895375d49f7e727efbcdc437053","last_reissued_at":"2026-05-18T03:09:09.238145Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:09.238145Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.12958","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":"2026-05-18T03:09:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZjUt2GT+I49ogl7HSaI50ht2hvaLR+C/W2q2A8YTy4H5T47xtgZEDXJN8RQQ+0YS3+sQit1f8feeN44t8mmeAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T00:46:55.540744Z"},"content_sha256":"aa7bd7e814bee4414845aec433fe99e71dcb6a9d34cc8a4e5b26c289361826e4","schema_version":"1.0","event_id":"sha256:aa7bd7e814bee4414845aec433fe99e71dcb6a9d34cc8a4e5b26c289361826e4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:BSJ7UVDKXDVTLBDGGC5FNX6CUG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Elementary spectral invariants and three-dimensional Reeb dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Elementary spectral invariants of contact three-manifolds suffice to prove some results on the existence and properties of Reeb periodic orbits.","cross_cats":["math.DS"],"primary_cat":"math.SG","authors_text":"Michael Hutchings","submitted_at":"2026-05-13T03:43:20Z","abstract_excerpt":"We survey various recent results on the existence and properties of periodic orbits of Reeb vector fields in three dimensions. We give an introduction to the \"elementary spectral invariants\" of contact three-manifolds, and we explain how they can be used to prove some of these results. (The remaining results can be proved using spectral invariants from embedded contact homology, of which the elementary spectral invariants are a simplification.) We then review the \"alternative ECH capacities\" of symplectic four-manifolds, and explain how these can be modified to define the elementary spectral i"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Elementary spectral invariants of contact three-manifolds can be used to prove some results on the existence and properties of periodic orbits of Reeb vector fields, and they are a simplification of spectral invariants from embedded contact homology.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the elementary spectral invariants, defined by modifying alternative ECH capacities, retain enough information from the full ECH theory to support the claimed proofs for some results while being strictly simpler.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Elementary spectral invariants simplify embedded contact homology spectral invariants for contact three-manifolds and can be used to prove some results on periodic orbits of Reeb vector fields, with the remaining results requiring the full ECH invariants.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Elementary spectral invariants of contact three-manifolds suffice to prove some results on the existence and properties of Reeb periodic orbits.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"16d6956fe646f7aaa65877f6fc06b2ae19418d9d0250fb0d0cdb667e036f2f41"},"source":{"id":"2605.12958","kind":"arxiv","version":1},"verdict":{"id":"9d164aec-ef83-4841-9da2-1030a9082f4f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T02:21:32.065349Z","strongest_claim":"Elementary spectral invariants of contact three-manifolds can be used to prove some results on the existence and properties of periodic orbits of Reeb vector fields, and they are a simplification of spectral invariants from embedded contact homology.","one_line_summary":"Elementary spectral invariants simplify embedded contact homology spectral invariants for contact three-manifolds and can be used to prove some results on periodic orbits of Reeb vector fields, with the remaining results requiring the full ECH invariants.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the elementary spectral invariants, defined by modifying alternative ECH capacities, retain enough information from the full ECH theory to support the claimed proofs for some results while being strictly simpler.","pith_extraction_headline":"Elementary spectral invariants of contact three-manifolds suffice to prove some results on the existence and properties of Reeb periodic orbits."},"references":{"count":75,"sample":[{"doi":"","year":2005,"title":"C. Abbas, K. Cieliebak and H. Hofer,The Weinstein conjecture for planar contact structures in dimension three, Comm. Math. Helv.80(2005), 771–793","work_id":"2f463edd-7d62-41f1-934a-a1a77e17d9a3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"P. Albers, H. Geiges, and K. Zehmisch,Pseudorotations of the2-disc and Reeb flows on the3-sphere, Ergodic Theory Dynam. Systems,42(2022), 402–436","work_id":"70e757da-55f7-4277-8ecb-21802c42ab1f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"Bangert,On the existence of closed geodesics on two-spheres, Internat","work_id":"b17b5819-65d2-4f39-94a3-f9976bd9e661","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Beiner,Infinite ECH capacities and Anosov flows, in preparation","work_id":"aa3857c3-101d-40c6-a459-fdf01b09cfa4","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"M. Borman, Y. Eliashberg, and E. Murphy,Existence and classification of overtwisted contact structures in all dimensions, Acta Math.215(2015), 281– 361","work_id":"61be18c7-4e8b-400a-988b-a605a33aaf9f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":75,"snapshot_sha256":"447b78de7eddc2a26be0227d69961151530dc396a60152d14dd6a9cc61533d0f","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"485161f79bd556e6c61402de37d7a3f57ea533e322f250b5f0d6e56f9638680b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"9d164aec-ef83-4841-9da2-1030a9082f4f"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:09:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jBbBUGo0KJhuh54B9NsDhCjExV5VPjw2GmRXsq9dGOGLdyNNaU9LZWBGxDAM98kgAmLZd4VAn2s4XqcsdOrxCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T00:46:55.542042Z"},"content_sha256":"abb6c96e3be7174fe984ade5c6ff3f59361e319628aa56846511df1fb0852f3b","schema_version":"1.0","event_id":"sha256:abb6c96e3be7174fe984ade5c6ff3f59361e319628aa56846511df1fb0852f3b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BSJ7UVDKXDVTLBDGGC5FNX6CUG/bundle.json","state_url":"https://pith.science/pith/BSJ7UVDKXDVTLBDGGC5FNX6CUG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BSJ7UVDKXDVTLBDGGC5FNX6CUG/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":"2026-06-10T00:46:55Z","links":{"resolver":"https://pith.science/pith/BSJ7UVDKXDVTLBDGGC5FNX6CUG","bundle":"https://pith.science/pith/BSJ7UVDKXDVTLBDGGC5FNX6CUG/bundle.json","state":"https://pith.science/pith/BSJ7UVDKXDVTLBDGGC5FNX6CUG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BSJ7UVDKXDVTLBDGGC5FNX6CUG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:BSJ7UVDKXDVTLBDGGC5FNX6CUG","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":"fc4f6427719e750d4d64af90a40d8929c48bb13fce8588b54d710b056cd21c03","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2026-05-13T03:43:20Z","title_canon_sha256":"3a458b1383cc54e28ce52fc3b74706e2fd862ce24f154c9f1e155beb215eac3f"},"schema_version":"1.0","source":{"id":"2605.12958","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12958","created_at":"2026-05-18T03:09:09Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12958v1","created_at":"2026-05-18T03:09:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12958","created_at":"2026-05-18T03:09:09Z"},{"alias_kind":"pith_short_12","alias_value":"BSJ7UVDKXDVT","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"BSJ7UVDKXDVTLBDG","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"BSJ7UVDK","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:abb6c96e3be7174fe984ade5c6ff3f59361e319628aa56846511df1fb0852f3b","target":"graph","created_at":"2026-05-18T03:09:09Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Elementary spectral invariants of contact three-manifolds can be used to prove some results on the existence and properties of periodic orbits of Reeb vector fields, and they are a simplification of spectral invariants from embedded contact homology."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the elementary spectral invariants, defined by modifying alternative ECH capacities, retain enough information from the full ECH theory to support the claimed proofs for some results while being strictly simpler."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Elementary spectral invariants simplify embedded contact homology spectral invariants for contact three-manifolds and can be used to prove some results on periodic orbits of Reeb vector fields, with the remaining results requiring the full ECH invariants."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Elementary spectral invariants of contact three-manifolds suffice to prove some results on the existence and properties of Reeb periodic orbits."}],"snapshot_sha256":"16d6956fe646f7aaa65877f6fc06b2ae19418d9d0250fb0d0cdb667e036f2f41"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"485161f79bd556e6c61402de37d7a3f57ea533e322f250b5f0d6e56f9638680b"},"paper":{"abstract_excerpt":"We survey various recent results on the existence and properties of periodic orbits of Reeb vector fields in three dimensions. We give an introduction to the \"elementary spectral invariants\" of contact three-manifolds, and we explain how they can be used to prove some of these results. (The remaining results can be proved using spectral invariants from embedded contact homology, of which the elementary spectral invariants are a simplification.) We then review the \"alternative ECH capacities\" of symplectic four-manifolds, and explain how these can be modified to define the elementary spectral i","authors_text":"Michael Hutchings","cross_cats":["math.DS"],"headline":"Elementary spectral invariants of contact three-manifolds suffice to prove some results on the existence and properties of Reeb periodic orbits.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2026-05-13T03:43:20Z","title":"Elementary spectral invariants and three-dimensional Reeb dynamics"},"references":{"count":75,"internal_anchors":0,"resolved_work":75,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"C. Abbas, K. Cieliebak and H. Hofer,The Weinstein conjecture for planar contact structures in dimension three, Comm. Math. Helv.80(2005), 771–793","work_id":"2f463edd-7d62-41f1-934a-a1a77e17d9a3","year":2005},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"P. Albers, H. Geiges, and K. Zehmisch,Pseudorotations of the2-disc and Reeb flows on the3-sphere, Ergodic Theory Dynam. Systems,42(2022), 402–436","work_id":"70e757da-55f7-4277-8ecb-21802c42ab1f","year":2022},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Bangert,On the existence of closed geodesics on two-spheres, Internat","work_id":"b17b5819-65d2-4f39-94a3-f9976bd9e661","year":1993},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Beiner,Infinite ECH capacities and Anosov flows, in preparation","work_id":"aa3857c3-101d-40c6-a459-fdf01b09cfa4","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"M. Borman, Y. Eliashberg, and E. Murphy,Existence and classification of overtwisted contact structures in all dimensions, Acta Math.215(2015), 281– 361","work_id":"61be18c7-4e8b-400a-988b-a605a33aaf9f","year":2015}],"snapshot_sha256":"447b78de7eddc2a26be0227d69961151530dc396a60152d14dd6a9cc61533d0f"},"source":{"id":"2605.12958","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T02:21:32.065349Z","id":"9d164aec-ef83-4841-9da2-1030a9082f4f","model_set":{"reader":"grok-4.3"},"one_line_summary":"Elementary spectral invariants simplify embedded contact homology spectral invariants for contact three-manifolds and can be used to prove some results on periodic orbits of Reeb vector fields, with the remaining results requiring the full ECH invariants.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Elementary spectral invariants of contact three-manifolds suffice to prove some results on the existence and properties of Reeb periodic orbits.","strongest_claim":"Elementary spectral invariants of contact three-manifolds can be used to prove some results on the existence and properties of periodic orbits of Reeb vector fields, and they are a simplification of spectral invariants from embedded contact homology.","weakest_assumption":"That the elementary spectral invariants, defined by modifying alternative ECH capacities, retain enough information from the full ECH theory to support the claimed proofs for some results while being strictly simpler."}},"verdict_id":"9d164aec-ef83-4841-9da2-1030a9082f4f"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:aa7bd7e814bee4414845aec433fe99e71dcb6a9d34cc8a4e5b26c289361826e4","target":"record","created_at":"2026-05-18T03:09:09Z","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":"fc4f6427719e750d4d64af90a40d8929c48bb13fce8588b54d710b056cd21c03","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2026-05-13T03:43:20Z","title_canon_sha256":"3a458b1383cc54e28ce52fc3b74706e2fd862ce24f154c9f1e155beb215eac3f"},"schema_version":"1.0","source":{"id":"2605.12958","kind":"arxiv","version":1}},"canonical_sha256":"0c93fa546ab8eb35846630ba56dfc2a193af8895375d49f7e727efbcdc437053","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c93fa546ab8eb35846630ba56dfc2a193af8895375d49f7e727efbcdc437053","first_computed_at":"2026-05-18T03:09:09.238145Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:09.238145Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7T01brJCsRA/rdvk4w0XnV9/pbfIxPf0Qp9UGpmPMRjQ+iZjrCUqOJwFfps8hD60EvX3qQeYMDlL1KQiK4HuBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:09.238944Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12958","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa7bd7e814bee4414845aec433fe99e71dcb6a9d34cc8a4e5b26c289361826e4","sha256:abb6c96e3be7174fe984ade5c6ff3f59361e319628aa56846511df1fb0852f3b"],"state_sha256":"05b2540ffbf2d0c11c74ffa68cfba7e10f49902ab5bef6df7d967210f9b015c0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iLgzzwH/e8ZF/aUwmz+1sg8vuGYCZuY3OC5GJgQNZ3Qwt/gwLWfUzFXtHbTzOwueCZI7jYyZv8r9I+T6tQEoCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T00:46:55.547466Z","bundle_sha256":"4684a4b52615a2020b4fa0df7ddbcd0a5753e5c2d6fd4a9d9d6a591ec31436a7"}}