{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6J74KHX23CEHCNO5HJMML5RTXP","short_pith_number":"pith:6J74KHX2","canonical_record":{"source":{"id":"1502.04414","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-16T03:25:01Z","cross_cats_sorted":[],"title_canon_sha256":"b41cbb61301d8d5b11a16e7750f704b56000298927e1ec905dfa1bbe2a89f9e0","abstract_canon_sha256":"d4aff663a4d9bd4e460b550b51ad1e8766df010b48318fb399a469494d789e4b"},"schema_version":"1.0"},"canonical_sha256":"f27fc51efad8887135dd3a58c5f633bbc6a9723e3c26a82b74f60662a14779ff","source":{"kind":"arxiv","id":"1502.04414","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.04414","created_at":"2026-05-18T02:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1502.04414v1","created_at":"2026-05-18T02:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.04414","created_at":"2026-05-18T02:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"6J74KHX23CEH","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6J74KHX23CEHCNO5","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6J74KHX2","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6J74KHX23CEHCNO5HJMML5RTXP","target":"record","payload":{"canonical_record":{"source":{"id":"1502.04414","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-16T03:25:01Z","cross_cats_sorted":[],"title_canon_sha256":"b41cbb61301d8d5b11a16e7750f704b56000298927e1ec905dfa1bbe2a89f9e0","abstract_canon_sha256":"d4aff663a4d9bd4e460b550b51ad1e8766df010b48318fb399a469494d789e4b"},"schema_version":"1.0"},"canonical_sha256":"f27fc51efad8887135dd3a58c5f633bbc6a9723e3c26a82b74f60662a14779ff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:59.816984Z","signature_b64":"4Gj5UmORmuAcekKWP7QYeJrzXbcEJueGaSanW0cAMf2IaYuhf4WaB2xAmSVTbS/SYwpKvrKwKEoHkm/yWeKHCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f27fc51efad8887135dd3a58c5f633bbc6a9723e3c26a82b74f60662a14779ff","last_reissued_at":"2026-05-18T02:26:59.816569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:59.816569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.04414","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":"2026-05-18T02:26:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vjuUjhQHLaqYuqMNMBzFuMWk4O1YmSbYvN8niEUh6IAT37unFMwhrj7nA8jC3kGMjD1TfbbIsWiKYQNJXCm8Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:35:00.820639Z"},"content_sha256":"7656f5b111d1c4827d311490d469c5d10f3a76a45548f20db43ec595d586f99c","schema_version":"1.0","event_id":"sha256:7656f5b111d1c4827d311490d469c5d10f3a76a45548f20db43ec595d586f99c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6J74KHX23CEHCNO5HJMML5RTXP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Excursion Probability of Certain Non-centered Smooth Gaussian Random Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dan Cheng","submitted_at":"2015-02-16T03:25:01Z","abstract_excerpt":"Let $X = \\{X(t): t\\in T \\}$ be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space $T$, and let $A_u(X,T) = \\{t\\in T: X(t)\\geq u\\}$ be the excursion set of $X$ exceeding level $u$. Under certain smoothness and regularity conditions, it is shown that, as $u\\to \\infty$, the excursion probability $\\mathbb{P}\\{\\sup_{t\\in T} X(t)\\ge u \\}$ can be approximated by the expected Euler characteristic of $A_u(X,T)$, denoted by $\\mathbb{E}\\{\\chi(A_u(X,T))\\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04414","kind":"arxiv","version":1},"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":"2026-05-18T02:26:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PjZZbSWLavexUsC/VyvGM9D7zyolK9BaSFe2+pvCFE1P7t0dWHaZ/qtzVJkki5veI5H0KBE8sLCx5ONLaRcDAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:35:00.820997Z"},"content_sha256":"2658d52e83781a8906389672b0c6d090785cd96f55553755fe2889fc992866c7","schema_version":"1.0","event_id":"sha256:2658d52e83781a8906389672b0c6d090785cd96f55553755fe2889fc992866c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6J74KHX23CEHCNO5HJMML5RTXP/bundle.json","state_url":"https://pith.science/pith/6J74KHX23CEHCNO5HJMML5RTXP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6J74KHX23CEHCNO5HJMML5RTXP/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":"2026-05-28T08:35:00Z","links":{"resolver":"https://pith.science/pith/6J74KHX23CEHCNO5HJMML5RTXP","bundle":"https://pith.science/pith/6J74KHX23CEHCNO5HJMML5RTXP/bundle.json","state":"https://pith.science/pith/6J74KHX23CEHCNO5HJMML5RTXP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6J74KHX23CEHCNO5HJMML5RTXP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6J74KHX23CEHCNO5HJMML5RTXP","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":"d4aff663a4d9bd4e460b550b51ad1e8766df010b48318fb399a469494d789e4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-16T03:25:01Z","title_canon_sha256":"b41cbb61301d8d5b11a16e7750f704b56000298927e1ec905dfa1bbe2a89f9e0"},"schema_version":"1.0","source":{"id":"1502.04414","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.04414","created_at":"2026-05-18T02:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1502.04414v1","created_at":"2026-05-18T02:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.04414","created_at":"2026-05-18T02:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"6J74KHX23CEH","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6J74KHX23CEHCNO5","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6J74KHX2","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:2658d52e83781a8906389672b0c6d090785cd96f55553755fe2889fc992866c7","target":"graph","created_at":"2026-05-18T02:26:59Z","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 $X = \\{X(t): t\\in T \\}$ be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space $T$, and let $A_u(X,T) = \\{t\\in T: X(t)\\geq u\\}$ be the excursion set of $X$ exceeding level $u$. Under certain smoothness and regularity conditions, it is shown that, as $u\\to \\infty$, the excursion probability $\\mathbb{P}\\{\\sup_{t\\in T} X(t)\\ge u \\}$ can be approximated by the expected Euler characteristic of $A_u(X,T)$, denoted by $\\mathbb{E}\\{\\chi(A_u(X,T))\\}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic f","authors_text":"Dan Cheng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-16T03:25:01Z","title":"Excursion Probability of Certain Non-centered Smooth Gaussian Random Fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.04414","kind":"arxiv","version":1},"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":"sha256:7656f5b111d1c4827d311490d469c5d10f3a76a45548f20db43ec595d586f99c","target":"record","created_at":"2026-05-18T02:26:59Z","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":"d4aff663a4d9bd4e460b550b51ad1e8766df010b48318fb399a469494d789e4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-16T03:25:01Z","title_canon_sha256":"b41cbb61301d8d5b11a16e7750f704b56000298927e1ec905dfa1bbe2a89f9e0"},"schema_version":"1.0","source":{"id":"1502.04414","kind":"arxiv","version":1}},"canonical_sha256":"f27fc51efad8887135dd3a58c5f633bbc6a9723e3c26a82b74f60662a14779ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f27fc51efad8887135dd3a58c5f633bbc6a9723e3c26a82b74f60662a14779ff","first_computed_at":"2026-05-18T02:26:59.816569Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:59.816569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4Gj5UmORmuAcekKWP7QYeJrzXbcEJueGaSanW0cAMf2IaYuhf4WaB2xAmSVTbS/SYwpKvrKwKEoHkm/yWeKHCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:59.816984Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.04414","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7656f5b111d1c4827d311490d469c5d10f3a76a45548f20db43ec595d586f99c","sha256:2658d52e83781a8906389672b0c6d090785cd96f55553755fe2889fc992866c7"],"state_sha256":"ca32f18e54364964311005d38f56dc986f4742b30fb8946b1da3f866461da0b4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r8ceJHgwaBN7nxl3/46jk28lQMANIU3XnCjCfFkP5xsPNzsylkTAuvabYBjkJHE4NnK5dj8RyaNorQA+eS86Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T08:35:00.823043Z","bundle_sha256":"c6b9809aa204efb396fbad3309d26a0add694274200872696fe731f6f72ec239"}}