{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:MPF3IP6F2TZIGGTTFMCQTG3F7J","short_pith_number":"pith:MPF3IP6F","canonical_record":{"source":{"id":"2607.19844","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-22T07:30:15Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"feeee16a5e428b8fc6e689f9641e7a543a5783d58c3e9d858a3d89d1f1731369","abstract_canon_sha256":"4125495670b01e401c3ba46f7f52b39d51588edfe39b5c96f2250c44a4a98129"},"schema_version":"1.0"},"canonical_sha256":"63cbb43fc5d4f2831a732b05099b65fa794815b59b55f02d58d3b55436490dae","source":{"kind":"arxiv","id":"2607.19844","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.19844","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"arxiv_version","alias_value":"2607.19844v1","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19844","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"pith_short_12","alias_value":"MPF3IP6F2TZI","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"pith_short_16","alias_value":"MPF3IP6F2TZIGGTT","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"pith_short_8","alias_value":"MPF3IP6F","created_at":"2026-07-23T01:24:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:MPF3IP6F2TZIGGTTFMCQTG3F7J","target":"record","payload":{"canonical_record":{"source":{"id":"2607.19844","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-22T07:30:15Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"feeee16a5e428b8fc6e689f9641e7a543a5783d58c3e9d858a3d89d1f1731369","abstract_canon_sha256":"4125495670b01e401c3ba46f7f52b39d51588edfe39b5c96f2250c44a4a98129"},"schema_version":"1.0"},"canonical_sha256":"63cbb43fc5d4f2831a732b05099b65fa794815b59b55f02d58d3b55436490dae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:24:03.270336Z","signature_b64":"zqI5njcB2vkSHZgn1EuxLFpqsZ3lu39HvR0WJaby9nd5qI5mjZDKYPXQ5acrq5RndyAJ2yMBiQ9Q6uSQERjbCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63cbb43fc5d4f2831a732b05099b65fa794815b59b55f02d58d3b55436490dae","last_reissued_at":"2026-07-23T01:24:03.269443Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:24:03.269443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.19844","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":"2026-07-23T01:24:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6VbQQtl/+3VM+SZasskswOIf0HZyLX+TFjR9PZ1H0tgZ0pNaGuiiGTdNrUwb91CCu7YGvoAmGAuSNLPOxEztCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T11:30:43.631446Z"},"content_sha256":"afb6885a2e8430e638df222c4c91c705e4ba6d134f3a8f088e8d8adab752b1df","schema_version":"1.0","event_id":"sha256:afb6885a2e8430e638df222c4c91c705e4ba6d134f3a8f088e8d8adab752b1df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:MPF3IP6F2TZIGGTTFMCQTG3F7J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Neven Elezovi\\'c","submitted_at":"2026-07-22T07:30:15Z","abstract_excerpt":"Let $X\\sim Bin(N,p)$ with $0<p<1$ and $q=1-p$, let $\\nu=\\lceil Np\\rceil$ be the first lattice point not below the mean, and let $r$ be a fixed integer. We determine the complete asymptotic expansion, in powers of $N^{-1}$, of the binomial mass $\\Pr\\{X=\\nu+r\\}$ -- a quotient of gamma functions -- uniformly for $p$ in compact subintervals of $(0,1)$ and for bounded $r$. Because the mean $Np$ is not a lattice point, the coefficients cannot be constants: they are Bernoulli polynomials evaluated at the oscillating fractional displacement $h_N=\\nu-Np\\in[0,1)$, and are given in closed form to all ord"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19844","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/2607.19844/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":"2026-07-23T01:24:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TqhDYbjzRA8dH7PKrzozomj2vyI4k0PPsqqKxOWILTuGlTLiy8UQK1Wu8S1k/fb7NFBhMnGaVye3NJIebDQHDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T11:30:43.632420Z"},"content_sha256":"f673869714b6e4f887a893c09d92df8c548c457b126ab274f9a3ae47bacfc257","schema_version":"1.0","event_id":"sha256:f673869714b6e4f887a893c09d92df8c548c457b126ab274f9a3ae47bacfc257"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/bundle.json","state_url":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/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":"2026-08-12T11:30:43Z","links":{"resolver":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J","bundle":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/bundle.json","state":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MPF3IP6F2TZIGGTTFMCQTG3F7J","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":"4125495670b01e401c3ba46f7f52b39d51588edfe39b5c96f2250c44a4a98129","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-22T07:30:15Z","title_canon_sha256":"feeee16a5e428b8fc6e689f9641e7a543a5783d58c3e9d858a3d89d1f1731369"},"schema_version":"1.0","source":{"id":"2607.19844","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.19844","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"arxiv_version","alias_value":"2607.19844v1","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19844","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"pith_short_12","alias_value":"MPF3IP6F2TZI","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"pith_short_16","alias_value":"MPF3IP6F2TZIGGTT","created_at":"2026-07-23T01:24:03Z"},{"alias_kind":"pith_short_8","alias_value":"MPF3IP6F","created_at":"2026-07-23T01:24:03Z"}],"graph_snapshots":[{"event_id":"sha256:f673869714b6e4f887a893c09d92df8c548c457b126ab274f9a3ae47bacfc257","target":"graph","created_at":"2026-07-23T01:24:03Z","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/2607.19844/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $X\\sim Bin(N,p)$ with $0<p<1$ and $q=1-p$, let $\\nu=\\lceil Np\\rceil$ be the first lattice point not below the mean, and let $r$ be a fixed integer. We determine the complete asymptotic expansion, in powers of $N^{-1}$, of the binomial mass $\\Pr\\{X=\\nu+r\\}$ -- a quotient of gamma functions -- uniformly for $p$ in compact subintervals of $(0,1)$ and for bounded $r$. Because the mean $Np$ is not a lattice point, the coefficients cannot be constants: they are Bernoulli polynomials evaluated at the oscillating fractional displacement $h_N=\\nu-Np\\in[0,1)$, and are given in closed form to all ord","authors_text":"Neven Elezovi\\'c","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-22T07:30:15Z","title":"Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19844","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:afb6885a2e8430e638df222c4c91c705e4ba6d134f3a8f088e8d8adab752b1df","target":"record","created_at":"2026-07-23T01:24:03Z","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":"4125495670b01e401c3ba46f7f52b39d51588edfe39b5c96f2250c44a4a98129","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-22T07:30:15Z","title_canon_sha256":"feeee16a5e428b8fc6e689f9641e7a543a5783d58c3e9d858a3d89d1f1731369"},"schema_version":"1.0","source":{"id":"2607.19844","kind":"arxiv","version":1}},"canonical_sha256":"63cbb43fc5d4f2831a732b05099b65fa794815b59b55f02d58d3b55436490dae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63cbb43fc5d4f2831a732b05099b65fa794815b59b55f02d58d3b55436490dae","first_computed_at":"2026-07-23T01:24:03.269443Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-23T01:24:03.269443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zqI5njcB2vkSHZgn1EuxLFpqsZ3lu39HvR0WJaby9nd5qI5mjZDKYPXQ5acrq5RndyAJ2yMBiQ9Q6uSQERjbCQ==","signature_status":"signed_v1","signed_at":"2026-07-23T01:24:03.270336Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.19844","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:afb6885a2e8430e638df222c4c91c705e4ba6d134f3a8f088e8d8adab752b1df","sha256:f673869714b6e4f887a893c09d92df8c548c457b126ab274f9a3ae47bacfc257"],"state_sha256":"54b823aedf6dc8be8bf7992a710463ea0eb5f83ab5055b298eb046644832ec4c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1XjgNFKzenF9dQN/Tar15qZEIykQ5bSej494Ua+zEEG0DfCmkafV/OoFuAdeQbkguyo2swfgSDGw/1aKSLloAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T11:30:43.643335Z","bundle_sha256":"45e7a26e85fe171995e61cd903f1d991140cb49fa8a1c9157917074a29a6bccd"}}