{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ESZQ7HR7EO7RMJYL3WU6U7GXFS","short_pith_number":"pith:ESZQ7HR7","canonical_record":{"source":{"id":"1307.0221","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-30T16:56:56Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"c1d7d9106173e0405f4807cb81810bea0167a842ed29e35993b11a7e599a406d","abstract_canon_sha256":"33e0f563edb0853db8417348a10e6c1fec39de7b0e2c5d9bfd93494b4fefb47f"},"schema_version":"1.0"},"canonical_sha256":"24b30f9e3f23bf16270bdda9ea7cd72ca58911d0f54c2f09e12669514745bfab","source":{"kind":"arxiv","id":"1307.0221","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0221","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0221v4","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0221","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"pith_short_12","alias_value":"ESZQ7HR7EO7R","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"ESZQ7HR7EO7RMJYL","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"ESZQ7HR7","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ESZQ7HR7EO7RMJYL3WU6U7GXFS","target":"record","payload":{"canonical_record":{"source":{"id":"1307.0221","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-30T16:56:56Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"c1d7d9106173e0405f4807cb81810bea0167a842ed29e35993b11a7e599a406d","abstract_canon_sha256":"33e0f563edb0853db8417348a10e6c1fec39de7b0e2c5d9bfd93494b4fefb47f"},"schema_version":"1.0"},"canonical_sha256":"24b30f9e3f23bf16270bdda9ea7cd72ca58911d0f54c2f09e12669514745bfab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:24.857175Z","signature_b64":"iqhhSZQJA7hwgQAbEqjqSreloyzRbTx1k1Qen+uuEaAdeEsC/PT6FImPIwA8hSTFyY04tPH7QouhxZ0x+8yaAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24b30f9e3f23bf16270bdda9ea7cd72ca58911d0f54c2f09e12669514745bfab","last_reissued_at":"2026-05-18T01:06:24.856681Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:24.856681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.0221","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:06:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kzh5bGHE6UlCVMjuqgEzc1nrBMmpEJgU0z9oNA4jwz/HPAG4QtCKfxvNmWsrS4mC1KNJjoziIDf/lvVmC/t+DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:19:38.013670Z"},"content_sha256":"69297a34f05b5210a804edfa3f2d0c4951c6612b866a75037b3dba0bad2e9f3a","schema_version":"1.0","event_id":"sha256:69297a34f05b5210a804edfa3f2d0c4951c6612b866a75037b3dba0bad2e9f3a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ESZQ7HR7EO7RMJYL3WU6U7GXFS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Beardwood-Halton-Hammersley Theorem for Stationary Ergodic Sequences: a Counterexample","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.PR","authors_text":"Alessandro Arlotto, J. Michael Steele","submitted_at":"2013-06-30T16:56:56Z","abstract_excerpt":"We construct a stationary ergodic process $X_1, X_2, \\ldots $ such that each $X_t$ has the uniform distribution on the unit square and the length $L_n$ of the shortest path through the points $X_1, X_2, \\ldots,X_n$ is not asymptotic to a constant times the square root of $n$. In other words, we show that the Beardwood, Halton and Hammersley theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0221","kind":"arxiv","version":4},"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":"2026-05-18T01:06:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sbZeWIQ/jQDRWg9R2DCCOQTI0kEBommcCu3kBkg1cmw6b+2/Xf/2xnc/HeCOSe91rfHh/mNRGEWKAc2p6y3bBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:19:38.014040Z"},"content_sha256":"102b41c9f761af3d8b3986f3dd8ba90760e8091248a2f18dfe5732ea81381350","schema_version":"1.0","event_id":"sha256:102b41c9f761af3d8b3986f3dd8ba90760e8091248a2f18dfe5732ea81381350"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ESZQ7HR7EO7RMJYL3WU6U7GXFS/bundle.json","state_url":"https://pith.science/pith/ESZQ7HR7EO7RMJYL3WU6U7GXFS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ESZQ7HR7EO7RMJYL3WU6U7GXFS/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":"2026-05-31T18:19:38Z","links":{"resolver":"https://pith.science/pith/ESZQ7HR7EO7RMJYL3WU6U7GXFS","bundle":"https://pith.science/pith/ESZQ7HR7EO7RMJYL3WU6U7GXFS/bundle.json","state":"https://pith.science/pith/ESZQ7HR7EO7RMJYL3WU6U7GXFS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ESZQ7HR7EO7RMJYL3WU6U7GXFS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ESZQ7HR7EO7RMJYL3WU6U7GXFS","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":"33e0f563edb0853db8417348a10e6c1fec39de7b0e2c5d9bfd93494b4fefb47f","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-30T16:56:56Z","title_canon_sha256":"c1d7d9106173e0405f4807cb81810bea0167a842ed29e35993b11a7e599a406d"},"schema_version":"1.0","source":{"id":"1307.0221","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0221","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0221v4","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0221","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"pith_short_12","alias_value":"ESZQ7HR7EO7R","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"ESZQ7HR7EO7RMJYL","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"ESZQ7HR7","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:102b41c9f761af3d8b3986f3dd8ba90760e8091248a2f18dfe5732ea81381350","target":"graph","created_at":"2026-05-18T01:06:24Z","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 construct a stationary ergodic process $X_1, X_2, \\ldots $ such that each $X_t$ has the uniform distribution on the unit square and the length $L_n$ of the shortest path through the points $X_1, X_2, \\ldots,X_n$ is not asymptotic to a constant times the square root of $n$. In other words, we show that the Beardwood, Halton and Hammersley theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions.","authors_text":"Alessandro Arlotto, J. Michael Steele","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-30T16:56:56Z","title":"Beardwood-Halton-Hammersley Theorem for Stationary Ergodic Sequences: a Counterexample"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0221","kind":"arxiv","version":4},"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":"sha256:69297a34f05b5210a804edfa3f2d0c4951c6612b866a75037b3dba0bad2e9f3a","target":"record","created_at":"2026-05-18T01:06:24Z","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":"33e0f563edb0853db8417348a10e6c1fec39de7b0e2c5d9bfd93494b4fefb47f","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-06-30T16:56:56Z","title_canon_sha256":"c1d7d9106173e0405f4807cb81810bea0167a842ed29e35993b11a7e599a406d"},"schema_version":"1.0","source":{"id":"1307.0221","kind":"arxiv","version":4}},"canonical_sha256":"24b30f9e3f23bf16270bdda9ea7cd72ca58911d0f54c2f09e12669514745bfab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"24b30f9e3f23bf16270bdda9ea7cd72ca58911d0f54c2f09e12669514745bfab","first_computed_at":"2026-05-18T01:06:24.856681Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:06:24.856681Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iqhhSZQJA7hwgQAbEqjqSreloyzRbTx1k1Qen+uuEaAdeEsC/PT6FImPIwA8hSTFyY04tPH7QouhxZ0x+8yaAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:06:24.857175Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.0221","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69297a34f05b5210a804edfa3f2d0c4951c6612b866a75037b3dba0bad2e9f3a","sha256:102b41c9f761af3d8b3986f3dd8ba90760e8091248a2f18dfe5732ea81381350"],"state_sha256":"3adf2a7306279aa950253b10bbb3e5fc5e24c25982b90cf64f4c00559a135c0f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SU5vJBbNGG64HmNdSuBa/6SIuqQroHxkIY7ntRjWkTrZFcAK+9w7BjtZsYEHcUssE0vgCQAQiwyavQ+r/OeABg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T18:19:38.016126Z","bundle_sha256":"99f81921e5e06aec0db274c3d5ead47b381f85d1ff9215aaf3b3b92cd0fd26c8"}}