{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:I65E6FFPVQDTJ2LXHFIMZETH6U","short_pith_number":"pith:I65E6FFP","canonical_record":{"source":{"id":"2607.14022","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-15T16:54:55Z","cross_cats_sorted":[],"title_canon_sha256":"099d83cc5cc1e2514e9c593bca70bd80452e7c8ccf1b166c7bbd4e2bf6bb103f","abstract_canon_sha256":"16a40a2e1b350ad6cbc28e3eb91275439dc7b7b4db26b3879370644546595e6d"},"schema_version":"1.0"},"canonical_sha256":"47ba4f14afac0734e9773950cc9267f53b5684f4c4f5b05097027bef429daf67","source":{"kind":"arxiv","id":"2607.14022","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.14022","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"arxiv_version","alias_value":"2607.14022v1","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.14022","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"pith_short_12","alias_value":"I65E6FFPVQDT","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"pith_short_16","alias_value":"I65E6FFPVQDTJ2LX","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"pith_short_8","alias_value":"I65E6FFP","created_at":"2026-07-16T01:23:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:I65E6FFPVQDTJ2LXHFIMZETH6U","target":"record","payload":{"canonical_record":{"source":{"id":"2607.14022","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-15T16:54:55Z","cross_cats_sorted":[],"title_canon_sha256":"099d83cc5cc1e2514e9c593bca70bd80452e7c8ccf1b166c7bbd4e2bf6bb103f","abstract_canon_sha256":"16a40a2e1b350ad6cbc28e3eb91275439dc7b7b4db26b3879370644546595e6d"},"schema_version":"1.0"},"canonical_sha256":"47ba4f14afac0734e9773950cc9267f53b5684f4c4f5b05097027bef429daf67","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-16T01:23:20.086868Z","signature_b64":"ErvHOem63d4zOe1G9GqnhJ3zpTQH81pEYKPHim8XkFqs4eEt/oYHy1JhuOSB3565awRllE3leO8kMnlyveqiDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47ba4f14afac0734e9773950cc9267f53b5684f4c4f5b05097027bef429daf67","last_reissued_at":"2026-07-16T01:23:20.086003Z","signature_status":"signed_v1","first_computed_at":"2026-07-16T01:23:20.086003Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.14022","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":"2026-07-16T01:23:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MDSkJWKgnbUjZ+wVeA9BjFPghg+5TbGoemhIgzQxTBHtWeUbE1EuIdKIhArFp+ib8zgVBWus5ihEYU6VtITNBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T06:34:21.669777Z"},"content_sha256":"9183a8409544ead445771bf4ad0050c79e0baaa0eb46181d5414876d48c17686","schema_version":"1.0","event_id":"sha256:9183a8409544ead445771bf4ad0050c79e0baaa0eb46181d5414876d48c17686"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:I65E6FFPVQDTJ2LXHFIMZETH6U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Supersaturation for Hypergraph-Weighted Independent Sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sam Spiro","submitted_at":"2026-07-15T16:54:55Z","abstract_excerpt":"Many extremal problems can be viewed as finding large independent sets in an auxiliary hypergraph. We propose a generalization of this by looking for ``large'' independent sets $I$ in a hypergraph $\\mathcal{F}$ where ``large'' is measured by how many edges $I$ induces in another hypergraph $\\mathcal{H}$ on the same vertex set as $\\mathcal{F}$. We prove general supersaturation results for such extremal problems motivated by the breakthrough work of Ferber, McKinley and Samotij on counting $F$-free graphs. As applications, we prove new supersaturation bounds for generalized Tur\\'an problems, as "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14022","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/2607.14022/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":"2026-07-16T01:23:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eRuFO4n3Oy74HYgncwOhBbwzoKIoutRYyJhsTr9KnwhOBEKCBpJAz1m82M1gdxTIpgmVny13EEHKfSnobTqtCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T06:34:21.670293Z"},"content_sha256":"4f638fe407a546e0e49cd61a13d3ea991a80caa1afb50ff6b9ae15f190a6926c","schema_version":"1.0","event_id":"sha256:4f638fe407a546e0e49cd61a13d3ea991a80caa1afb50ff6b9ae15f190a6926c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I65E6FFPVQDTJ2LXHFIMZETH6U/bundle.json","state_url":"https://pith.science/pith/I65E6FFPVQDTJ2LXHFIMZETH6U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I65E6FFPVQDTJ2LXHFIMZETH6U/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":"2026-08-09T06:34:21Z","links":{"resolver":"https://pith.science/pith/I65E6FFPVQDTJ2LXHFIMZETH6U","bundle":"https://pith.science/pith/I65E6FFPVQDTJ2LXHFIMZETH6U/bundle.json","state":"https://pith.science/pith/I65E6FFPVQDTJ2LXHFIMZETH6U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I65E6FFPVQDTJ2LXHFIMZETH6U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:I65E6FFPVQDTJ2LXHFIMZETH6U","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":"16a40a2e1b350ad6cbc28e3eb91275439dc7b7b4db26b3879370644546595e6d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-15T16:54:55Z","title_canon_sha256":"099d83cc5cc1e2514e9c593bca70bd80452e7c8ccf1b166c7bbd4e2bf6bb103f"},"schema_version":"1.0","source":{"id":"2607.14022","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.14022","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"arxiv_version","alias_value":"2607.14022v1","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.14022","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"pith_short_12","alias_value":"I65E6FFPVQDT","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"pith_short_16","alias_value":"I65E6FFPVQDTJ2LX","created_at":"2026-07-16T01:23:20Z"},{"alias_kind":"pith_short_8","alias_value":"I65E6FFP","created_at":"2026-07-16T01:23:20Z"}],"graph_snapshots":[{"event_id":"sha256:4f638fe407a546e0e49cd61a13d3ea991a80caa1afb50ff6b9ae15f190a6926c","target":"graph","created_at":"2026-07-16T01:23:20Z","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/2607.14022/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Many extremal problems can be viewed as finding large independent sets in an auxiliary hypergraph. We propose a generalization of this by looking for ``large'' independent sets $I$ in a hypergraph $\\mathcal{F}$ where ``large'' is measured by how many edges $I$ induces in another hypergraph $\\mathcal{H}$ on the same vertex set as $\\mathcal{F}$. We prove general supersaturation results for such extremal problems motivated by the breakthrough work of Ferber, McKinley and Samotij on counting $F$-free graphs. As applications, we prove new supersaturation bounds for generalized Tur\\'an problems, as ","authors_text":"Sam Spiro","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-15T16:54:55Z","title":"Supersaturation for Hypergraph-Weighted Independent Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14022","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":"sha256:9183a8409544ead445771bf4ad0050c79e0baaa0eb46181d5414876d48c17686","target":"record","created_at":"2026-07-16T01:23:20Z","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":"16a40a2e1b350ad6cbc28e3eb91275439dc7b7b4db26b3879370644546595e6d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-15T16:54:55Z","title_canon_sha256":"099d83cc5cc1e2514e9c593bca70bd80452e7c8ccf1b166c7bbd4e2bf6bb103f"},"schema_version":"1.0","source":{"id":"2607.14022","kind":"arxiv","version":1}},"canonical_sha256":"47ba4f14afac0734e9773950cc9267f53b5684f4c4f5b05097027bef429daf67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47ba4f14afac0734e9773950cc9267f53b5684f4c4f5b05097027bef429daf67","first_computed_at":"2026-07-16T01:23:20.086003Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-16T01:23:20.086003Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ErvHOem63d4zOe1G9GqnhJ3zpTQH81pEYKPHim8XkFqs4eEt/oYHy1JhuOSB3565awRllE3leO8kMnlyveqiDQ==","signature_status":"signed_v1","signed_at":"2026-07-16T01:23:20.086868Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.14022","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9183a8409544ead445771bf4ad0050c79e0baaa0eb46181d5414876d48c17686","sha256:4f638fe407a546e0e49cd61a13d3ea991a80caa1afb50ff6b9ae15f190a6926c"],"state_sha256":"e95f520ac9a0c0369072dbb00be03c7d29077dac963b0bce62f4a964725b4d01"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zk3jw5uol8PaDnIEC747ycyadSAUcuNvSKTnfE7ROXCsPW2b+aab58UQ4H//cQF97rdbflysPQhWLknYs6w8Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T06:34:21.675364Z","bundle_sha256":"e7a8628a95787f4d4dd39bdfc10d2390299166ab6abc51aaccafbd98f435c070"}}