{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:EGSUFBVI6FNXWL4JWCQKOHCF5R","short_pith_number":"pith:EGSUFBVI","canonical_record":{"source":{"id":"1509.06191","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2015-09-21T11:31:01Z","cross_cats_sorted":[],"title_canon_sha256":"9ea1aae4fba898ec8d6b396a130c9c3e3068eeed98846eb1bd0f67b6c5de64c5","abstract_canon_sha256":"dc584e5b32bd6e312f3d5aa284d1295fcb56c7bf475c396faff69eb1ea772de3"},"schema_version":"1.0"},"canonical_sha256":"21a54286a8f15b7b2f89b0a0a71c45ec59fed852d7180a43374aa75aa19cb81e","source":{"kind":"arxiv","id":"1509.06191","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.06191","created_at":"2026-05-17T23:57:33Z"},{"alias_kind":"arxiv_version","alias_value":"1509.06191v3","created_at":"2026-05-17T23:57:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06191","created_at":"2026-05-17T23:57:33Z"},{"alias_kind":"pith_short_12","alias_value":"EGSUFBVI6FNX","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"EGSUFBVI6FNXWL4J","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"EGSUFBVI","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:EGSUFBVI6FNXWL4JWCQKOHCF5R","target":"record","payload":{"canonical_record":{"source":{"id":"1509.06191","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2015-09-21T11:31:01Z","cross_cats_sorted":[],"title_canon_sha256":"9ea1aae4fba898ec8d6b396a130c9c3e3068eeed98846eb1bd0f67b6c5de64c5","abstract_canon_sha256":"dc584e5b32bd6e312f3d5aa284d1295fcb56c7bf475c396faff69eb1ea772de3"},"schema_version":"1.0"},"canonical_sha256":"21a54286a8f15b7b2f89b0a0a71c45ec59fed852d7180a43374aa75aa19cb81e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:33.251806Z","signature_b64":"Z2HFo6zxD1ts8sqYCSS/I1DBsxf/hNmPv3C1s+MsiqHMqERPdLF5acsJLI5PrCCD+TvazQ4LSsISLTwRAYb6Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21a54286a8f15b7b2f89b0a0a71c45ec59fed852d7180a43374aa75aa19cb81e","last_reissued_at":"2026-05-17T23:57:33.251342Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:33.251342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.06191","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:57:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JrD5bMEaMcngjARXlYFChpgXdMTV3LShEHOOHtJQOJU/Ev2DHhjyYf3VdM7RBtaNmL43MQd+YAUnKaRJFtvPBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:22:05.600423Z"},"content_sha256":"2c43ccb5c3a0ae614658a69d98204b1ed45247bebf4446e754e15afeff84024c","schema_version":"1.0","event_id":"sha256:2c43ccb5c3a0ae614658a69d98204b1ed45247bebf4446e754e15afeff84024c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:EGSUFBVI6FNXWL4JWCQKOHCF5R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Elchanan Mossel, Jan H\\k{a}z{\\l}a, Thomas Holenstein","submitted_at":"2015-09-21T11:31:01Z","abstract_excerpt":"There is a common theme to some research questions in additive combinatorics and noise stability. Both study the following basic question: Let $\\mathcal{P}$ be a probability distribution over a space $\\Omega^\\ell$ with all $\\ell$ marginals equal. Let $\\underline{X}^{(1)}, \\ldots, \\underline{X}^{(\\ell)}$ where $\\underline{X}^{(j)} = (X_1^{(j)}, \\ldots, X_n^{(j)})$ be random vectors such that for every coordinate $i \\in [n]$ the tuples $(X_i^{(1)}, \\ldots, X_i^{(\\ell)})$ are i.i.d. according to $\\mathcal{P}$.\n  A central question that is addressed in both areas is:\n  - Does there exist a functio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06191","kind":"arxiv","version":3},"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":"2026-05-17T23:57:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RjBov7lt+rrLD8P9T7G7oPgJiPghpv3ctBzWj51cU/fqbFEnH3TPMOKO+DejKQBlWT7NIF7mgVuaL4ZZGLkFDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T03:22:05.601066Z"},"content_sha256":"96c373bbbd1857df5186b93a6aa1fb1dad7ef5265be4deb002d23191b3bc8ee9","schema_version":"1.0","event_id":"sha256:96c373bbbd1857df5186b93a6aa1fb1dad7ef5265be4deb002d23191b3bc8ee9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EGSUFBVI6FNXWL4JWCQKOHCF5R/bundle.json","state_url":"https://pith.science/pith/EGSUFBVI6FNXWL4JWCQKOHCF5R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EGSUFBVI6FNXWL4JWCQKOHCF5R/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":"2026-05-31T03:22:05Z","links":{"resolver":"https://pith.science/pith/EGSUFBVI6FNXWL4JWCQKOHCF5R","bundle":"https://pith.science/pith/EGSUFBVI6FNXWL4JWCQKOHCF5R/bundle.json","state":"https://pith.science/pith/EGSUFBVI6FNXWL4JWCQKOHCF5R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EGSUFBVI6FNXWL4JWCQKOHCF5R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:EGSUFBVI6FNXWL4JWCQKOHCF5R","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":"dc584e5b32bd6e312f3d5aa284d1295fcb56c7bf475c396faff69eb1ea772de3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2015-09-21T11:31:01Z","title_canon_sha256":"9ea1aae4fba898ec8d6b396a130c9c3e3068eeed98846eb1bd0f67b6c5de64c5"},"schema_version":"1.0","source":{"id":"1509.06191","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.06191","created_at":"2026-05-17T23:57:33Z"},{"alias_kind":"arxiv_version","alias_value":"1509.06191v3","created_at":"2026-05-17T23:57:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06191","created_at":"2026-05-17T23:57:33Z"},{"alias_kind":"pith_short_12","alias_value":"EGSUFBVI6FNX","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"EGSUFBVI6FNXWL4J","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"EGSUFBVI","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:96c373bbbd1857df5186b93a6aa1fb1dad7ef5265be4deb002d23191b3bc8ee9","target":"graph","created_at":"2026-05-17T23:57:33Z","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":"There is a common theme to some research questions in additive combinatorics and noise stability. Both study the following basic question: Let $\\mathcal{P}$ be a probability distribution over a space $\\Omega^\\ell$ with all $\\ell$ marginals equal. Let $\\underline{X}^{(1)}, \\ldots, \\underline{X}^{(\\ell)}$ where $\\underline{X}^{(j)} = (X_1^{(j)}, \\ldots, X_n^{(j)})$ be random vectors such that for every coordinate $i \\in [n]$ the tuples $(X_i^{(1)}, \\ldots, X_i^{(\\ell)})$ are i.i.d. according to $\\mathcal{P}$.\n  A central question that is addressed in both areas is:\n  - Does there exist a functio","authors_text":"Elchanan Mossel, Jan H\\k{a}z{\\l}a, Thomas Holenstein","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2015-09-21T11:31:01Z","title":"Product Space Models of Correlation: Between Noise Stability and Additive Combinatorics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06191","kind":"arxiv","version":3},"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":"sha256:2c43ccb5c3a0ae614658a69d98204b1ed45247bebf4446e754e15afeff84024c","target":"record","created_at":"2026-05-17T23:57:33Z","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":"dc584e5b32bd6e312f3d5aa284d1295fcb56c7bf475c396faff69eb1ea772de3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2015-09-21T11:31:01Z","title_canon_sha256":"9ea1aae4fba898ec8d6b396a130c9c3e3068eeed98846eb1bd0f67b6c5de64c5"},"schema_version":"1.0","source":{"id":"1509.06191","kind":"arxiv","version":3}},"canonical_sha256":"21a54286a8f15b7b2f89b0a0a71c45ec59fed852d7180a43374aa75aa19cb81e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"21a54286a8f15b7b2f89b0a0a71c45ec59fed852d7180a43374aa75aa19cb81e","first_computed_at":"2026-05-17T23:57:33.251342Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:33.251342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z2HFo6zxD1ts8sqYCSS/I1DBsxf/hNmPv3C1s+MsiqHMqERPdLF5acsJLI5PrCCD+TvazQ4LSsISLTwRAYb6Bw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:33.251806Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.06191","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c43ccb5c3a0ae614658a69d98204b1ed45247bebf4446e754e15afeff84024c","sha256:96c373bbbd1857df5186b93a6aa1fb1dad7ef5265be4deb002d23191b3bc8ee9"],"state_sha256":"c027839742a526fdb9f31ba9eade4291a948aad405fbe2bb7874e6366e0895a7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dq7Z5TTjiuEsKQpjytmmYdWTNkQcn+BQtnNSRM+kaF1Dm5JNrF0tc/T/KC4ShfrzI36HgebCeMU6wtnUiAhiDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T03:22:05.604461Z","bundle_sha256":"5555e2d021e9da1e2b99d96cdc4c8a1d625459a3b2068510b17c55ab2e9f6d9b"}}