{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:4J6KHA723L5UVYKWGI7MCAJ5QR","short_pith_number":"pith:4J6KHA72","canonical_record":{"source":{"id":"1111.2060","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-08T21:17:51Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"3773f33248340f8c2aaacacab86837bc1473c34379e64092231379caf4e1b578","abstract_canon_sha256":"37eb6e7a1b776b33e5a4f19b8c27148efb8421593616862b3c4b3f403f884ef1"},"schema_version":"1.0"},"canonical_sha256":"e27ca383fadafb4ae156323ec1013d8446b7439bf78d1af7ab48886454e1d32e","source":{"kind":"arxiv","id":"1111.2060","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2060","created_at":"2026-05-18T02:29:32Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2060v2","created_at":"2026-05-18T02:29:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2060","created_at":"2026-05-18T02:29:32Z"},{"alias_kind":"pith_short_12","alias_value":"4J6KHA723L5U","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4J6KHA723L5UVYKW","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4J6KHA72","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:4J6KHA723L5UVYKWGI7MCAJ5QR","target":"record","payload":{"canonical_record":{"source":{"id":"1111.2060","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-08T21:17:51Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"3773f33248340f8c2aaacacab86837bc1473c34379e64092231379caf4e1b578","abstract_canon_sha256":"37eb6e7a1b776b33e5a4f19b8c27148efb8421593616862b3c4b3f403f884ef1"},"schema_version":"1.0"},"canonical_sha256":"e27ca383fadafb4ae156323ec1013d8446b7439bf78d1af7ab48886454e1d32e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:32.684358Z","signature_b64":"9i0Pm9B0gjr61rKk5xame3Y0evOR3/bgiHuYVU/5mIh6+3/ka7TjdXKiAQcQsTXicraqeLr0kjr/Lf1da5pYBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e27ca383fadafb4ae156323ec1013d8446b7439bf78d1af7ab48886454e1d32e","last_reissued_at":"2026-05-18T02:29:32.683952Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:32.683952Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.2060","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":"2026-05-18T02:29:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OopZ72yeIxO6gM5A4bGRYDOzenMjvVboJVGGhvWKXhxUnIwJP0Lov4i6enM5tMVqJimmyDVOwmlHVnTMxjRhDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:23:24.889007Z"},"content_sha256":"fb0c62fc301dc98158c78c3444bd73b313125d83484da91f960efd20dae1936b","schema_version":"1.0","event_id":"sha256:fb0c62fc301dc98158c78c3444bd73b313125d83484da91f960efd20dae1936b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:4J6KHA723L5UVYKWGI7MCAJ5QR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DS","authors_text":"Steven P. Lalley","submitted_at":"2011-11-08T21:17:51Z","abstract_excerpt":"Let $\\Upsilon $ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $\\Upsilon$ of length $\\leq L$, choose one, say $\\gamma_{L}$, at random and let $N (\\gamma_{L})$ be the number of its self-intersections. It is known that there is a positive constant $\\kappa$ depending on the metric such that $N (\\gamma_{L})/L^{2} \\rightarrow \\kappa$ in probability as $L\\rightarrow \\infty$. The main results of this paper concern the size of typical fluctuations of $N (\\gamma_{L})$ about $\\kappa L^{2}$. It is proved that if the metric has constant curvature -1 then typical "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2060","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":"2026-05-18T02:29:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H+wO4lgZNvXvlBGC+K+BRdIWqmuxv7E8Y6P3cb8Z3p0ycDjrj0AGom49/NIBISpmSt/zABR360dXByC6+PUEAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:23:24.889369Z"},"content_sha256":"7dd27ea3c1d65758bc8ef23bc464f56cd387f9d663c46528e0a5945173da84e7","schema_version":"1.0","event_id":"sha256:7dd27ea3c1d65758bc8ef23bc464f56cd387f9d663c46528e0a5945173da84e7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4J6KHA723L5UVYKWGI7MCAJ5QR/bundle.json","state_url":"https://pith.science/pith/4J6KHA723L5UVYKWGI7MCAJ5QR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4J6KHA723L5UVYKWGI7MCAJ5QR/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":"2026-06-02T12:23:24Z","links":{"resolver":"https://pith.science/pith/4J6KHA723L5UVYKWGI7MCAJ5QR","bundle":"https://pith.science/pith/4J6KHA723L5UVYKWGI7MCAJ5QR/bundle.json","state":"https://pith.science/pith/4J6KHA723L5UVYKWGI7MCAJ5QR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4J6KHA723L5UVYKWGI7MCAJ5QR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:4J6KHA723L5UVYKWGI7MCAJ5QR","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":"37eb6e7a1b776b33e5a4f19b8c27148efb8421593616862b3c4b3f403f884ef1","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-08T21:17:51Z","title_canon_sha256":"3773f33248340f8c2aaacacab86837bc1473c34379e64092231379caf4e1b578"},"schema_version":"1.0","source":{"id":"1111.2060","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2060","created_at":"2026-05-18T02:29:32Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2060v2","created_at":"2026-05-18T02:29:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2060","created_at":"2026-05-18T02:29:32Z"},{"alias_kind":"pith_short_12","alias_value":"4J6KHA723L5U","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4J6KHA723L5UVYKW","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4J6KHA72","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:7dd27ea3c1d65758bc8ef23bc464f56cd387f9d663c46528e0a5945173da84e7","target":"graph","created_at":"2026-05-18T02:29:32Z","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 $\\Upsilon $ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $\\Upsilon$ of length $\\leq L$, choose one, say $\\gamma_{L}$, at random and let $N (\\gamma_{L})$ be the number of its self-intersections. It is known that there is a positive constant $\\kappa$ depending on the metric such that $N (\\gamma_{L})/L^{2} \\rightarrow \\kappa$ in probability as $L\\rightarrow \\infty$. The main results of this paper concern the size of typical fluctuations of $N (\\gamma_{L})$ about $\\kappa L^{2}$. It is proved that if the metric has constant curvature -1 then typical ","authors_text":"Steven P. Lalley","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-08T21:17:51Z","title":"Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2060","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:fb0c62fc301dc98158c78c3444bd73b313125d83484da91f960efd20dae1936b","target":"record","created_at":"2026-05-18T02:29:32Z","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":"37eb6e7a1b776b33e5a4f19b8c27148efb8421593616862b3c4b3f403f884ef1","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-11-08T21:17:51Z","title_canon_sha256":"3773f33248340f8c2aaacacab86837bc1473c34379e64092231379caf4e1b578"},"schema_version":"1.0","source":{"id":"1111.2060","kind":"arxiv","version":2}},"canonical_sha256":"e27ca383fadafb4ae156323ec1013d8446b7439bf78d1af7ab48886454e1d32e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e27ca383fadafb4ae156323ec1013d8446b7439bf78d1af7ab48886454e1d32e","first_computed_at":"2026-05-18T02:29:32.683952Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:32.683952Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9i0Pm9B0gjr61rKk5xame3Y0evOR3/bgiHuYVU/5mIh6+3/ka7TjdXKiAQcQsTXicraqeLr0kjr/Lf1da5pYBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:32.684358Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.2060","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb0c62fc301dc98158c78c3444bd73b313125d83484da91f960efd20dae1936b","sha256:7dd27ea3c1d65758bc8ef23bc464f56cd387f9d663c46528e0a5945173da84e7"],"state_sha256":"0d116dcfdd45e5e830f1a11b414a868931dcffa06f0ea42410f60afe550064be"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vwtzjWchrx0xCbMG3PcRvJoAj1DhOWd4nvEkllUBGDfc3azl4KS+1Fsb5bLBVm5FPdm+nAeS36RXYxg+bVHCAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T12:23:24.891279Z","bundle_sha256":"f3a7b3c6147bccd35de7fe141edb664cc8fc826f8194ce8cbc21c0855a853052"}}