{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:DVSHGOSLNMBCKMAYEXQPWKUZJ6","short_pith_number":"pith:DVSHGOSL","canonical_record":{"source":{"id":"2606.11139","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T17:29:46Z","cross_cats_sorted":[],"title_canon_sha256":"6e3ea76135d6813697981c0897b37bcef6a16021fcc58841a57933b449338ebf","abstract_canon_sha256":"9d9c21d5d38a884e2a4cbf7d2989d7e6167f43ec2794ddb8b0bdeb3106e917ea"},"schema_version":"1.0"},"canonical_sha256":"1d64733a4b6b0225301825e0fb2a994fbf011bb6d7908391724baf35f06cdfdf","source":{"kind":"arxiv","id":"2606.11139","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11139","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11139v1","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11139","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_12","alias_value":"DVSHGOSLNMBC","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_16","alias_value":"DVSHGOSLNMBCKMAY","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_8","alias_value":"DVSHGOSL","created_at":"2026-06-10T01:11:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:DVSHGOSLNMBCKMAYEXQPWKUZJ6","target":"record","payload":{"canonical_record":{"source":{"id":"2606.11139","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T17:29:46Z","cross_cats_sorted":[],"title_canon_sha256":"6e3ea76135d6813697981c0897b37bcef6a16021fcc58841a57933b449338ebf","abstract_canon_sha256":"9d9c21d5d38a884e2a4cbf7d2989d7e6167f43ec2794ddb8b0bdeb3106e917ea"},"schema_version":"1.0"},"canonical_sha256":"1d64733a4b6b0225301825e0fb2a994fbf011bb6d7908391724baf35f06cdfdf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:11:13.164449Z","signature_b64":"noffvODaoxzC6o76igzNmeOI3ugC8AnNApEh3SiQR2VTz+z7Q0YXBVyNCDC+xinsG1ch492EmWDmKMwq0CRYCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d64733a4b6b0225301825e0fb2a994fbf011bb6d7908391724baf35f06cdfdf","last_reissued_at":"2026-06-10T01:11:13.163814Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:11:13.163814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.11139","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":"2026-06-10T01:11:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Txm4QAMApghxRJP5cxgBTnViQLIRhZLZaVKtWtRuLUATkxMLtwcDy4IqsCNIIAqV7GOQJ01si7lc0jJu/iKVBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:36:19.465402Z"},"content_sha256":"f6da24ebce86de64cd00818b3123be6e67d947e6367d5b7249c9374312cfbca8","schema_version":"1.0","event_id":"sha256:f6da24ebce86de64cd00818b3123be6e67d947e6367d5b7249c9374312cfbca8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:DVSHGOSLNMBCKMAYEXQPWKUZJ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp bounds on $k$-wise generalizations of oddtowns and eventowns","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lan Wei, Minghui Ouyang, Zichao Dong","submitted_at":"2026-06-09T17:29:46Z","abstract_excerpt":"For $\\boldsymbol{\\alpha} = (\\alpha_1, \\dots, \\alpha_k) \\in \\mathbb{F}_2^k$, an $\\boldsymbol{\\alpha}$-town is a set family in which every $i$-wise intersection has parity $\\alpha_i$. Denote by $f_{\\boldsymbol{\\alpha}}(n)$ the maximum size of an $\\boldsymbol{\\alpha}$-town on $[n]$. The classical oddtown and eventown problems study the cases $\\boldsymbol{\\alpha} = (1, 0)$ and $(0, 0)$, respectively. We determine the sharp asymptotics of $f_{\\boldsymbol{\\alpha}}(n)$ for all $\\boldsymbol{\\alpha}$, answering questions of Johnston--O'Neill and Wei--Zhang--Ge.\n  We also study a symmetric variant $g_{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11139","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/2606.11139/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":"2026-06-10T01:11:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4ND65VEx8mJSy7sJ+V7RrulZGtNGiPMuxG4r9Jf+oteJAYhiSr+Q6x+R8cPybDfHWr69JsoZHxhZCQt2+4MsAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:36:19.466167Z"},"content_sha256":"e1d78331076490243b36d50c0468a56b90bf16d1beef996686758b4d165b9b8c","schema_version":"1.0","event_id":"sha256:e1d78331076490243b36d50c0468a56b90bf16d1beef996686758b4d165b9b8c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DVSHGOSLNMBCKMAYEXQPWKUZJ6/bundle.json","state_url":"https://pith.science/pith/DVSHGOSLNMBCKMAYEXQPWKUZJ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DVSHGOSLNMBCKMAYEXQPWKUZJ6/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":"2026-06-10T21:36:19Z","links":{"resolver":"https://pith.science/pith/DVSHGOSLNMBCKMAYEXQPWKUZJ6","bundle":"https://pith.science/pith/DVSHGOSLNMBCKMAYEXQPWKUZJ6/bundle.json","state":"https://pith.science/pith/DVSHGOSLNMBCKMAYEXQPWKUZJ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DVSHGOSLNMBCKMAYEXQPWKUZJ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DVSHGOSLNMBCKMAYEXQPWKUZJ6","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":"9d9c21d5d38a884e2a4cbf7d2989d7e6167f43ec2794ddb8b0bdeb3106e917ea","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T17:29:46Z","title_canon_sha256":"6e3ea76135d6813697981c0897b37bcef6a16021fcc58841a57933b449338ebf"},"schema_version":"1.0","source":{"id":"2606.11139","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11139","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11139v1","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11139","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_12","alias_value":"DVSHGOSLNMBC","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_16","alias_value":"DVSHGOSLNMBCKMAY","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_8","alias_value":"DVSHGOSL","created_at":"2026-06-10T01:11:13Z"}],"graph_snapshots":[{"event_id":"sha256:e1d78331076490243b36d50c0468a56b90bf16d1beef996686758b4d165b9b8c","target":"graph","created_at":"2026-06-10T01:11:13Z","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/2606.11139/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For $\\boldsymbol{\\alpha} = (\\alpha_1, \\dots, \\alpha_k) \\in \\mathbb{F}_2^k$, an $\\boldsymbol{\\alpha}$-town is a set family in which every $i$-wise intersection has parity $\\alpha_i$. Denote by $f_{\\boldsymbol{\\alpha}}(n)$ the maximum size of an $\\boldsymbol{\\alpha}$-town on $[n]$. The classical oddtown and eventown problems study the cases $\\boldsymbol{\\alpha} = (1, 0)$ and $(0, 0)$, respectively. We determine the sharp asymptotics of $f_{\\boldsymbol{\\alpha}}(n)$ for all $\\boldsymbol{\\alpha}$, answering questions of Johnston--O'Neill and Wei--Zhang--Ge.\n  We also study a symmetric variant $g_{\\","authors_text":"Lan Wei, Minghui Ouyang, Zichao Dong","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T17:29:46Z","title":"Sharp bounds on $k$-wise generalizations of oddtowns and eventowns"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11139","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:f6da24ebce86de64cd00818b3123be6e67d947e6367d5b7249c9374312cfbca8","target":"record","created_at":"2026-06-10T01:11:13Z","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":"9d9c21d5d38a884e2a4cbf7d2989d7e6167f43ec2794ddb8b0bdeb3106e917ea","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T17:29:46Z","title_canon_sha256":"6e3ea76135d6813697981c0897b37bcef6a16021fcc58841a57933b449338ebf"},"schema_version":"1.0","source":{"id":"2606.11139","kind":"arxiv","version":1}},"canonical_sha256":"1d64733a4b6b0225301825e0fb2a994fbf011bb6d7908391724baf35f06cdfdf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d64733a4b6b0225301825e0fb2a994fbf011bb6d7908391724baf35f06cdfdf","first_computed_at":"2026-06-10T01:11:13.163814Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:11:13.163814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"noffvODaoxzC6o76igzNmeOI3ugC8AnNApEh3SiQR2VTz+z7Q0YXBVyNCDC+xinsG1ch492EmWDmKMwq0CRYCw==","signature_status":"signed_v1","signed_at":"2026-06-10T01:11:13.164449Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11139","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f6da24ebce86de64cd00818b3123be6e67d947e6367d5b7249c9374312cfbca8","sha256:e1d78331076490243b36d50c0468a56b90bf16d1beef996686758b4d165b9b8c"],"state_sha256":"17a5622693c818a5ca5db496b32bb7a40fc64ec10d70864ef5b965a0ac9a466d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kjt71qGlYr7ra6nPIouADb0bAGYfZYT2RnK3UlVWEloRuUWNJn3yYXgSNcX8blIA57KnhhKm/uQcmwLnSy8dDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T21:36:19.470392Z","bundle_sha256":"510b463dd1646555e434f4bdefbdb69737a421f66a9bd07cfc24c9b1f4544b6f"}}