{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ZUO2F4GV3GD642CYZLKBKQGM6C","short_pith_number":"pith:ZUO2F4GV","canonical_record":{"source":{"id":"2607.08415","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.ST","submitted_at":"2026-07-09T12:39:29Z","cross_cats_sorted":["math.PR","stat.TH"],"title_canon_sha256":"61814e13b184dd74bbf989fbc2c8b6bd2f57f024cfd2521bc49478b0da126836","abstract_canon_sha256":"058a71c688fe9aa28c683437fca3ab923f216afdce9d438984340d51c2029098"},"schema_version":"1.0"},"canonical_sha256":"cd1da2f0d5d987ee6858cad41540ccf0bba000ffb217cbbe41d1be72bcc7cbba","source":{"kind":"arxiv","id":"2607.08415","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.08415","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"arxiv_version","alias_value":"2607.08415v1","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.08415","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"pith_short_12","alias_value":"ZUO2F4GV3GD6","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"pith_short_16","alias_value":"ZUO2F4GV3GD642CY","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"pith_short_8","alias_value":"ZUO2F4GV","created_at":"2026-07-10T01:19:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ZUO2F4GV3GD642CYZLKBKQGM6C","target":"record","payload":{"canonical_record":{"source":{"id":"2607.08415","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.ST","submitted_at":"2026-07-09T12:39:29Z","cross_cats_sorted":["math.PR","stat.TH"],"title_canon_sha256":"61814e13b184dd74bbf989fbc2c8b6bd2f57f024cfd2521bc49478b0da126836","abstract_canon_sha256":"058a71c688fe9aa28c683437fca3ab923f216afdce9d438984340d51c2029098"},"schema_version":"1.0"},"canonical_sha256":"cd1da2f0d5d987ee6858cad41540ccf0bba000ffb217cbbe41d1be72bcc7cbba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-10T01:19:49.916440Z","signature_b64":"MXH4GneEylfHTJHgmwQE8qqxp43EE8Ue5DRz4D93TXeUiG+ihi+v0ZcZvXebgFtwui0Qef+CZMq8wXdOc5X8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd1da2f0d5d987ee6858cad41540ccf0bba000ffb217cbbe41d1be72bcc7cbba","last_reissued_at":"2026-07-10T01:19:49.916008Z","signature_status":"signed_v1","first_computed_at":"2026-07-10T01:19:49.916008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.08415","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":"2026-07-10T01:19:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D65o8o/gwyNjSCK2UQQeD0WM83dQ2cydKK9Q+0a3OYm+k4b136uwL2BTxLMUf3Hw54RqAnw728KR98tvtBoACg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-02T10:24:07.308365Z"},"content_sha256":"cb8ed4b9b76c33d5da2e39229f6fa3ee19e945cb9ea1a9802f143fd5355d98f2","schema_version":"1.0","event_id":"sha256:cb8ed4b9b76c33d5da2e39229f6fa3ee19e945cb9ea1a9802f143fd5355d98f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ZUO2F4GV3GD642CYZLKBKQGM6C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Exact Distribution-Free Test for Means of Nonnegative Random Variables","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Nikos Vlassis, Philip S. Thomas","submitted_at":"2026-07-09T12:39:29Z","abstract_excerpt":"Let $X=(X_1,\\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\\ldots,D_n)\\sim\\operatorname{Dir}(1,\\ldots,1)$ be independent of $X$, and define $K(x)=\\mathbb{P}\\{\\sum_{i=1}^n x_iD_i\\le1\\}$. We prove that, for every $n\\ge1$, whenever $\\mathbb{E} X_i\\le1$ for every $i$, $\\mathbb{P}\\{K(X)\\le\\alpha\\}\\le\\alpha$ for all $0\\le\\alpha\\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\\mathbb{E}X_i \\le 1$ for all $i$. This proves a conjecture of Gaffke (2005)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.08415","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.08415/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-10T01:19:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X7xsIwx27au6I2XHZ8KH84fTSIyW4ys3nzqHzPO8meb3IATLq752C+qh61bo/FlFVSi4xB4KnucUiqGAxSqXAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-02T10:24:07.309037Z"},"content_sha256":"9d538b3c2a3d3bdef059322eec45f22d8461088dada9400788b8cc73a625062d","schema_version":"1.0","event_id":"sha256:9d538b3c2a3d3bdef059322eec45f22d8461088dada9400788b8cc73a625062d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZUO2F4GV3GD642CYZLKBKQGM6C/bundle.json","state_url":"https://pith.science/pith/ZUO2F4GV3GD642CYZLKBKQGM6C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZUO2F4GV3GD642CYZLKBKQGM6C/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":"2026-08-02T10:24:07Z","links":{"resolver":"https://pith.science/pith/ZUO2F4GV3GD642CYZLKBKQGM6C","bundle":"https://pith.science/pith/ZUO2F4GV3GD642CYZLKBKQGM6C/bundle.json","state":"https://pith.science/pith/ZUO2F4GV3GD642CYZLKBKQGM6C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZUO2F4GV3GD642CYZLKBKQGM6C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ZUO2F4GV3GD642CYZLKBKQGM6C","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":"058a71c688fe9aa28c683437fca3ab923f216afdce9d438984340d51c2029098","cross_cats_sorted":["math.PR","stat.TH"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.ST","submitted_at":"2026-07-09T12:39:29Z","title_canon_sha256":"61814e13b184dd74bbf989fbc2c8b6bd2f57f024cfd2521bc49478b0da126836"},"schema_version":"1.0","source":{"id":"2607.08415","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.08415","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"arxiv_version","alias_value":"2607.08415v1","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.08415","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"pith_short_12","alias_value":"ZUO2F4GV3GD6","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"pith_short_16","alias_value":"ZUO2F4GV3GD642CY","created_at":"2026-07-10T01:19:49Z"},{"alias_kind":"pith_short_8","alias_value":"ZUO2F4GV","created_at":"2026-07-10T01:19:49Z"}],"graph_snapshots":[{"event_id":"sha256:9d538b3c2a3d3bdef059322eec45f22d8461088dada9400788b8cc73a625062d","target":"graph","created_at":"2026-07-10T01:19:49Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.08415/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $X=(X_1,\\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\\ldots,D_n)\\sim\\operatorname{Dir}(1,\\ldots,1)$ be independent of $X$, and define $K(x)=\\mathbb{P}\\{\\sum_{i=1}^n x_iD_i\\le1\\}$. We prove that, for every $n\\ge1$, whenever $\\mathbb{E} X_i\\le1$ for every $i$, $\\mathbb{P}\\{K(X)\\le\\alpha\\}\\le\\alpha$ for all $0\\le\\alpha\\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\\mathbb{E}X_i \\le 1$ for all $i$. This proves a conjecture of Gaffke (2005).","authors_text":"Nikos Vlassis, Philip S. Thomas","cross_cats":["math.PR","stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.ST","submitted_at":"2026-07-09T12:39:29Z","title":"An Exact Distribution-Free Test for Means of Nonnegative Random Variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.08415","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:cb8ed4b9b76c33d5da2e39229f6fa3ee19e945cb9ea1a9802f143fd5355d98f2","target":"record","created_at":"2026-07-10T01:19:49Z","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":"058a71c688fe9aa28c683437fca3ab923f216afdce9d438984340d51c2029098","cross_cats_sorted":["math.PR","stat.TH"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.ST","submitted_at":"2026-07-09T12:39:29Z","title_canon_sha256":"61814e13b184dd74bbf989fbc2c8b6bd2f57f024cfd2521bc49478b0da126836"},"schema_version":"1.0","source":{"id":"2607.08415","kind":"arxiv","version":1}},"canonical_sha256":"cd1da2f0d5d987ee6858cad41540ccf0bba000ffb217cbbe41d1be72bcc7cbba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd1da2f0d5d987ee6858cad41540ccf0bba000ffb217cbbe41d1be72bcc7cbba","first_computed_at":"2026-07-10T01:19:49.916008Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-10T01:19:49.916008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MXH4GneEylfHTJHgmwQE8qqxp43EE8Ue5DRz4D93TXeUiG+ihi+v0ZcZvXebgFtwui0Qef+CZMq8wXdOc5X8BQ==","signature_status":"signed_v1","signed_at":"2026-07-10T01:19:49.916440Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.08415","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb8ed4b9b76c33d5da2e39229f6fa3ee19e945cb9ea1a9802f143fd5355d98f2","sha256:9d538b3c2a3d3bdef059322eec45f22d8461088dada9400788b8cc73a625062d"],"state_sha256":"fed12fb4dfe54cde1c02cb2de133871da568a171e778341d77840aa1370020be"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LTiTqTM/ZkHogpDzvatnauG5MdQ0m8ES4PG81NjhY8oClc00VFvnNJpg7kKVGyu0I2IsKgZjRwKtHGCOTAxIDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-02T10:24:07.313042Z","bundle_sha256":"23c9daa90f6a634def5a1cf7ba8ce1210b013c894860bfb923677774c45239fa"}}