{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5GZ7464TE4QKSDZQ6FRHHRKDT5","short_pith_number":"pith:5GZ7464T","canonical_record":{"source":{"id":"2605.17157","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-16T21:01:43Z","cross_cats_sorted":[],"title_canon_sha256":"af43c5fd122888ceb2820cb45562871d4ad088af77767d7971507e948c5556c8","abstract_canon_sha256":"f191643cab07173bdf433f766124640176bee8ce59d199549dbc4b0f5a287cb1"},"schema_version":"1.0"},"canonical_sha256":"e9b3fe7b932720a90f30f16273c5439f4b2f273fdba511246e4d6c8902f9ef75","source":{"kind":"arxiv","id":"2605.17157","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.17157","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"arxiv_version","alias_value":"2605.17157v1","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17157","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"pith_short_12","alias_value":"5GZ7464TE4QK","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"pith_short_16","alias_value":"5GZ7464TE4QKSDZQ","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"pith_short_8","alias_value":"5GZ7464T","created_at":"2026-05-20T00:03:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5GZ7464TE4QKSDZQ6FRHHRKDT5","target":"record","payload":{"canonical_record":{"source":{"id":"2605.17157","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-16T21:01:43Z","cross_cats_sorted":[],"title_canon_sha256":"af43c5fd122888ceb2820cb45562871d4ad088af77767d7971507e948c5556c8","abstract_canon_sha256":"f191643cab07173bdf433f766124640176bee8ce59d199549dbc4b0f5a287cb1"},"schema_version":"1.0"},"canonical_sha256":"e9b3fe7b932720a90f30f16273c5439f4b2f273fdba511246e4d6c8902f9ef75","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:03:42.541052Z","signature_b64":"FYNNRySZ5OEKOCnWX6bSSgVQq6oMkyCWEeosNRga2A1Pa8sPfs62sh3yQkBj8llSlaCXRUnwD9e7Jg8fHwVSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9b3fe7b932720a90f30f16273c5439f4b2f273fdba511246e4d6c8902f9ef75","last_reissued_at":"2026-05-20T00:03:42.540229Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:03:42.540229Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.17157","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":"2026-05-20T00:03:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zI0u9nVv+dGCm4SNX2rrNXht2j440u951cTzC82ABRjssQeYaaxUZJcay8tNoo79h/Det8pbn6qnAmNaM4HxCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T16:58:44.619745Z"},"content_sha256":"131356ce6d80e16649860a0fdb23ac43e75ee43663d213bb73b497f87dc67a52","schema_version":"1.0","event_id":"sha256:131356ce6d80e16649860a0fdb23ac43e75ee43663d213bb73b497f87dc67a52"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5GZ7464TE4QKSDZQ6FRHHRKDT5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gaps of Binary Numerical Semigroups and of Binary Inclusion-Exclusion Polynomials","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gennady Bachman","submitted_at":"2026-05-16T21:01:43Z","abstract_excerpt":"Let $p$ be a given modulus, let $u$ be prime to $p$, and consider the linear permutation $u\\cdot n\\pmod p$ of the residue system modulo $p$. Writing $\\langle x\\rangle_p$ to denote the least nonnegative residue of $x$ modulo $p$, we say that a pair of integers $(a,b)$ is a dominant pair of this permutation if either the inequality $\\max(\\langle ua\\rangle_p,\\langle ub\\rangle_p)<\\min_{a<n<b}\\langle un\\rangle_p$, or the inequality $\\min(\\langle ua\\rangle_p,\\langle ub\\rangle_p)>\\max_{a<n<b}\\langle un\\rangle_p$ hold. The main technical part of this work gives analysis of this property of linear perm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17157","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/2605.17157/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T22:33:23.761859Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.992358Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"00316de25f2098a2906e2fb6ac5aad824c35408abb7ac0da3707762e3a204039"},"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":"2026-05-20T00:03:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HktgiBkkgRjWa/cqQnauhEqoXI5RZ6T325ltBvq9mgeuMAZ/G4tbZSCq8AJMxtosxUeBL5M5/H9dBwRHsRC0Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T16:58:44.620500Z"},"content_sha256":"a432dde4b62334b91a55ec0208a7fe574a721c863922ff90e16e4d73517ca256","schema_version":"1.0","event_id":"sha256:a432dde4b62334b91a55ec0208a7fe574a721c863922ff90e16e4d73517ca256"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5GZ7464TE4QKSDZQ6FRHHRKDT5/bundle.json","state_url":"https://pith.science/pith/5GZ7464TE4QKSDZQ6FRHHRKDT5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5GZ7464TE4QKSDZQ6FRHHRKDT5/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":"2026-05-30T16:58:44Z","links":{"resolver":"https://pith.science/pith/5GZ7464TE4QKSDZQ6FRHHRKDT5","bundle":"https://pith.science/pith/5GZ7464TE4QKSDZQ6FRHHRKDT5/bundle.json","state":"https://pith.science/pith/5GZ7464TE4QKSDZQ6FRHHRKDT5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5GZ7464TE4QKSDZQ6FRHHRKDT5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5GZ7464TE4QKSDZQ6FRHHRKDT5","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":"f191643cab07173bdf433f766124640176bee8ce59d199549dbc4b0f5a287cb1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-16T21:01:43Z","title_canon_sha256":"af43c5fd122888ceb2820cb45562871d4ad088af77767d7971507e948c5556c8"},"schema_version":"1.0","source":{"id":"2605.17157","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.17157","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"arxiv_version","alias_value":"2605.17157v1","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17157","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"pith_short_12","alias_value":"5GZ7464TE4QK","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"pith_short_16","alias_value":"5GZ7464TE4QKSDZQ","created_at":"2026-05-20T00:03:42Z"},{"alias_kind":"pith_short_8","alias_value":"5GZ7464T","created_at":"2026-05-20T00:03:42Z"}],"graph_snapshots":[{"event_id":"sha256:a432dde4b62334b91a55ec0208a7fe574a721c863922ff90e16e4d73517ca256","target":"graph","created_at":"2026-05-20T00:03:42Z","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":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-19T22:33:23.761859Z","status":"skipped","version":"1.0.0"},{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.992358Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.17157/integrity.json","findings":[],"snapshot_sha256":"00316de25f2098a2906e2fb6ac5aad824c35408abb7ac0da3707762e3a204039","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $p$ be a given modulus, let $u$ be prime to $p$, and consider the linear permutation $u\\cdot n\\pmod p$ of the residue system modulo $p$. Writing $\\langle x\\rangle_p$ to denote the least nonnegative residue of $x$ modulo $p$, we say that a pair of integers $(a,b)$ is a dominant pair of this permutation if either the inequality $\\max(\\langle ua\\rangle_p,\\langle ub\\rangle_p)<\\min_{a<n<b}\\langle un\\rangle_p$, or the inequality $\\min(\\langle ua\\rangle_p,\\langle ub\\rangle_p)>\\max_{a<n<b}\\langle un\\rangle_p$ hold. The main technical part of this work gives analysis of this property of linear perm","authors_text":"Gennady Bachman","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-16T21:01:43Z","title":"Gaps of Binary Numerical Semigroups and of Binary Inclusion-Exclusion Polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17157","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":"sha256:131356ce6d80e16649860a0fdb23ac43e75ee43663d213bb73b497f87dc67a52","target":"record","created_at":"2026-05-20T00:03:42Z","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":"f191643cab07173bdf433f766124640176bee8ce59d199549dbc4b0f5a287cb1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-16T21:01:43Z","title_canon_sha256":"af43c5fd122888ceb2820cb45562871d4ad088af77767d7971507e948c5556c8"},"schema_version":"1.0","source":{"id":"2605.17157","kind":"arxiv","version":1}},"canonical_sha256":"e9b3fe7b932720a90f30f16273c5439f4b2f273fdba511246e4d6c8902f9ef75","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e9b3fe7b932720a90f30f16273c5439f4b2f273fdba511246e4d6c8902f9ef75","first_computed_at":"2026-05-20T00:03:42.540229Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:42.540229Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FYNNRySZ5OEKOCnWX6bSSgVQq6oMkyCWEeosNRga2A1Pa8sPfs62sh3yQkBj8llSlaCXRUnwD9e7Jg8fHwVSBg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:42.541052Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17157","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:131356ce6d80e16649860a0fdb23ac43e75ee43663d213bb73b497f87dc67a52","sha256:a432dde4b62334b91a55ec0208a7fe574a721c863922ff90e16e4d73517ca256"],"state_sha256":"ee7a158c2411eb26aa8ff81bb364bd25b42717e32152a2333c81dbc213cd53c0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uZBEpDcq8mFzjrXYmDgXG4LsmxPfsw5o3U5xywGPuUrxmX7FlVx+Zjbi/kO6hCzDQTMNfT1UfBHUfvT3KdhuDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T16:58:44.623570Z","bundle_sha256":"bf293a10dd933f20c8af2eb0e5d4ec7a9798aeb91b96d04c33a3ffbddb5faa58"}}