{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3NZFI2W2ASLKVCMVUOYMGFODNJ","short_pith_number":"pith:3NZFI2W2","canonical_record":{"source":{"id":"2607.06443","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T16:10:29Z","cross_cats_sorted":[],"title_canon_sha256":"69b58b00cf40efe5ec3d7f4291bf76d66bb5ed0f4ebc9acc3f3eb7536a1f6326","abstract_canon_sha256":"b12899b0694c7d0bab3314b0166d4c7634f4176623d98a4ae97279b1b6bb1c86"},"schema_version":"1.0"},"canonical_sha256":"db72546ada0496aa8995a3b0c315c36a4d1097f3d4e772e9447c94f43b95f159","source":{"kind":"arxiv","id":"2607.06443","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.06443","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"arxiv_version","alias_value":"2607.06443v1","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06443","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"pith_short_12","alias_value":"3NZFI2W2ASLK","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"pith_short_16","alias_value":"3NZFI2W2ASLKVCMV","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"pith_short_8","alias_value":"3NZFI2W2","created_at":"2026-07-08T01:19:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3NZFI2W2ASLKVCMVUOYMGFODNJ","target":"record","payload":{"canonical_record":{"source":{"id":"2607.06443","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T16:10:29Z","cross_cats_sorted":[],"title_canon_sha256":"69b58b00cf40efe5ec3d7f4291bf76d66bb5ed0f4ebc9acc3f3eb7536a1f6326","abstract_canon_sha256":"b12899b0694c7d0bab3314b0166d4c7634f4176623d98a4ae97279b1b6bb1c86"},"schema_version":"1.0"},"canonical_sha256":"db72546ada0496aa8995a3b0c315c36a4d1097f3d4e772e9447c94f43b95f159","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:19:25.418670Z","signature_b64":"7mIPKmBPU3QHTWIMvaOOf7QKXmj22GYs1NemLnpdvkxi0KzuJh3Z83cMEVmblfXbYPsKyGCkoIwp7NKlXKORAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db72546ada0496aa8995a3b0c315c36a4d1097f3d4e772e9447c94f43b95f159","last_reissued_at":"2026-07-08T01:19:25.418241Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:19:25.418241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.06443","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":"2026-07-08T01:19:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"606VA7Se1hBOYQasd3BqveyfMVnX3ghQeQzbmCsf+6DTJL6odsMr4blJy2+tgBSpV9OIt+ANEyu2y1j9eOS5BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T17:41:03.368755Z"},"content_sha256":"637f30ef6cb6715429c8ae81a2cbf07adbc2723cffaa8c7c1861eb51529067b6","schema_version":"1.0","event_id":"sha256:637f30ef6cb6715429c8ae81a2cbf07adbc2723cffaa8c7c1861eb51529067b6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3NZFI2W2ASLKVCMVUOYMGFODNJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a conjecture regarding the product version of the Hilton-Milner theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lihua Feng, Lu Lu, Qifan Wang, Xucheng Bu, Zejun Huang","submitted_at":"2026-07-07T16:10:29Z","abstract_excerpt":"Recently, Frankl and Wang considered a product version of the classical Hilton-Milner theorem. They conjectured that, if $\\mathcal{F} \\subset \\binom{[n]}{k}$ and $\\mathcal{G} \\subset \\binom{[n]}{\\ell}$ are non-trivial cross-intersecting families with $n \\geq 2k > 2\\ell \\geq 4$, the maximum of $|\\mathcal{F}||\\mathcal{G}|$ is attained by the natural Hilton-Milner-type configurations.\n  In this paper, we present two main results concerning this conjecture. Firstly, we show that the conjecture does not hold in general. By introducing a two-center construction, we prove that for every fixed integer"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06443","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/2607.06443/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":"2026-07-08T01:19:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aMIKK6bllmstDaeYTJ6vM9JwxLQrGZEgsjw9IPQecAzwU12tkGCj7fkxn+ObdLTNGXrvmzTd54Ca5zHuzxyqCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T17:41:03.369252Z"},"content_sha256":"062a7541b81fd48db0a2883a53510880731f9f28db9cf48641fed7ae63b73d16","schema_version":"1.0","event_id":"sha256:062a7541b81fd48db0a2883a53510880731f9f28db9cf48641fed7ae63b73d16"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3NZFI2W2ASLKVCMVUOYMGFODNJ/bundle.json","state_url":"https://pith.science/pith/3NZFI2W2ASLKVCMVUOYMGFODNJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3NZFI2W2ASLKVCMVUOYMGFODNJ/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":"2026-08-05T17:41:03Z","links":{"resolver":"https://pith.science/pith/3NZFI2W2ASLKVCMVUOYMGFODNJ","bundle":"https://pith.science/pith/3NZFI2W2ASLKVCMVUOYMGFODNJ/bundle.json","state":"https://pith.science/pith/3NZFI2W2ASLKVCMVUOYMGFODNJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3NZFI2W2ASLKVCMVUOYMGFODNJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3NZFI2W2ASLKVCMVUOYMGFODNJ","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":"b12899b0694c7d0bab3314b0166d4c7634f4176623d98a4ae97279b1b6bb1c86","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T16:10:29Z","title_canon_sha256":"69b58b00cf40efe5ec3d7f4291bf76d66bb5ed0f4ebc9acc3f3eb7536a1f6326"},"schema_version":"1.0","source":{"id":"2607.06443","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.06443","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"arxiv_version","alias_value":"2607.06443v1","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06443","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"pith_short_12","alias_value":"3NZFI2W2ASLK","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"pith_short_16","alias_value":"3NZFI2W2ASLKVCMV","created_at":"2026-07-08T01:19:25Z"},{"alias_kind":"pith_short_8","alias_value":"3NZFI2W2","created_at":"2026-07-08T01:19:25Z"}],"graph_snapshots":[{"event_id":"sha256:062a7541b81fd48db0a2883a53510880731f9f28db9cf48641fed7ae63b73d16","target":"graph","created_at":"2026-07-08T01:19:25Z","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/2607.06443/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Recently, Frankl and Wang considered a product version of the classical Hilton-Milner theorem. They conjectured that, if $\\mathcal{F} \\subset \\binom{[n]}{k}$ and $\\mathcal{G} \\subset \\binom{[n]}{\\ell}$ are non-trivial cross-intersecting families with $n \\geq 2k > 2\\ell \\geq 4$, the maximum of $|\\mathcal{F}||\\mathcal{G}|$ is attained by the natural Hilton-Milner-type configurations.\n  In this paper, we present two main results concerning this conjecture. Firstly, we show that the conjecture does not hold in general. By introducing a two-center construction, we prove that for every fixed integer","authors_text":"Lihua Feng, Lu Lu, Qifan Wang, Xucheng Bu, Zejun Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T16:10:29Z","title":"On a conjecture regarding the product version of the Hilton-Milner theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06443","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":"sha256:637f30ef6cb6715429c8ae81a2cbf07adbc2723cffaa8c7c1861eb51529067b6","target":"record","created_at":"2026-07-08T01:19:25Z","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":"b12899b0694c7d0bab3314b0166d4c7634f4176623d98a4ae97279b1b6bb1c86","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T16:10:29Z","title_canon_sha256":"69b58b00cf40efe5ec3d7f4291bf76d66bb5ed0f4ebc9acc3f3eb7536a1f6326"},"schema_version":"1.0","source":{"id":"2607.06443","kind":"arxiv","version":1}},"canonical_sha256":"db72546ada0496aa8995a3b0c315c36a4d1097f3d4e772e9447c94f43b95f159","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db72546ada0496aa8995a3b0c315c36a4d1097f3d4e772e9447c94f43b95f159","first_computed_at":"2026-07-08T01:19:25.418241Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-08T01:19:25.418241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7mIPKmBPU3QHTWIMvaOOf7QKXmj22GYs1NemLnpdvkxi0KzuJh3Z83cMEVmblfXbYPsKyGCkoIwp7NKlXKORAw==","signature_status":"signed_v1","signed_at":"2026-07-08T01:19:25.418670Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.06443","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:637f30ef6cb6715429c8ae81a2cbf07adbc2723cffaa8c7c1861eb51529067b6","sha256:062a7541b81fd48db0a2883a53510880731f9f28db9cf48641fed7ae63b73d16"],"state_sha256":"8106e145e5560482479fe99d1047db92006cbe27ae3d4d57c88432606b99bac0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nGNlrWC/PDuuWr8nY783FSTLWGVM0fDyKK5+B97usqFOBz3aNXawDGGsl6N3pJKq3RDklEm1NQmF4SFZQDeMBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T17:41:03.372866Z","bundle_sha256":"995091fc9a334a4cc3d3aba182a8ce79f5184508537b4f0e46e89d0c5ff5fac0"}}