{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:SSGON2DJYHUUOLM5NVCB5YFPG5","short_pith_number":"pith:SSGON2DJ","canonical_record":{"source":{"id":"2507.06730","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-09T10:43:11Z","cross_cats_sorted":[],"title_canon_sha256":"031763eac56d75acc71d0fd7e0d9074e678afb70f25ee82f9b2681e64f574985","abstract_canon_sha256":"0011f26ccf390400bb1b4305986523d9beece881379712564c89deeab85e170e"},"schema_version":"1.0"},"canonical_sha256":"948ce6e869c1e9472d9d6d441ee0af3775d565ccdb8c032833ef6ba8f6221dae","source":{"kind":"arxiv","id":"2507.06730","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06730","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06730v1","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06730","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"pith_short_12","alias_value":"SSGON2DJYHUU","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"pith_short_16","alias_value":"SSGON2DJYHUUOLM5","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"pith_short_8","alias_value":"SSGON2DJ","created_at":"2026-07-05T11:34:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:SSGON2DJYHUUOLM5NVCB5YFPG5","target":"record","payload":{"canonical_record":{"source":{"id":"2507.06730","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-09T10:43:11Z","cross_cats_sorted":[],"title_canon_sha256":"031763eac56d75acc71d0fd7e0d9074e678afb70f25ee82f9b2681e64f574985","abstract_canon_sha256":"0011f26ccf390400bb1b4305986523d9beece881379712564c89deeab85e170e"},"schema_version":"1.0"},"canonical_sha256":"948ce6e869c1e9472d9d6d441ee0af3775d565ccdb8c032833ef6ba8f6221dae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:34:22.169083Z","signature_b64":"1Wc0PcEfP0p8Xaq7an+IFLNSEi7IU8v/S8tLpkfOC0mQ6DmoQ7xTpXbzjOPqfEwprujCfTzRr39RHSMHTHQhBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"948ce6e869c1e9472d9d6d441ee0af3775d565ccdb8c032833ef6ba8f6221dae","last_reissued_at":"2026-07-05T11:34:22.168667Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:34:22.168667Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.06730","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":"2026-07-05T11:34:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k50aaB6qFu3RRLK+CRW88YoiZXWpBbaFgWOx2vocYukihY+biDyzYurcIDhlrMYjTshsMbQ6YdIYYX35FawOAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:51:01.306727Z"},"content_sha256":"1a340874ad45b5d8d2078a4495c21b2da98bf7fb406f73eff2654edbad8126c0","schema_version":"1.0","event_id":"sha256:1a340874ad45b5d8d2078a4495c21b2da98bf7fb406f73eff2654edbad8126c0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:SSGON2DJYHUUOLM5NVCB5YFPG5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Sierpi\\'{n}ski packing chromatic number and recognition of Sierpi\\'{n}ski products","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"P\\v{r}emysl Holub, Sandi Klav\\v{z}ar","submitted_at":"2025-07-09T10:43:11Z","abstract_excerpt":"The Sierpi\\'{n}ski product $G \\otimes _f H$ of graphs $G$ and $H$ with respect to a function $f \\colon V(G)\\rightarrow V(H)$ has the vertex set $V(G)\\times V(H)$. For every $g\\in V(G)$ it contains a disjoint copy $gH$ of $H$, and for every edge $gg'$ of $G$ there is the edge $(g,f(g'))(g',f(g))$ between $gH$ and $g'H$. In this paper, the Sierpi\\'{n}ski packing chromatic number is defined as the minimum of $\\chi_{\\rho}(G\\otimes _f H)$ over all functions $f$, where $\\chi_{\\rho}(X)$ is the packing chromatic number of $X$. The upper Sierpi\\'{n}ski packing chromatic number is analogously defined as"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06730","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/2507.06730/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":"2026-07-05T11:34:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2fTpoiojRHiJ/YGef7+c4paC/HEYMbnNxJTLeN31UMkUwI1oC0o52B+wr5Au8uITrD36h5RoHVbl2diCYmgoAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T11:51:01.307302Z"},"content_sha256":"ba84db50867739c77331bda338afb29dd52756ffb476b7bba4b7eb3f97195bc2","schema_version":"1.0","event_id":"sha256:ba84db50867739c77331bda338afb29dd52756ffb476b7bba4b7eb3f97195bc2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SSGON2DJYHUUOLM5NVCB5YFPG5/bundle.json","state_url":"https://pith.science/pith/SSGON2DJYHUUOLM5NVCB5YFPG5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SSGON2DJYHUUOLM5NVCB5YFPG5/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":"2026-08-13T11:51:01Z","links":{"resolver":"https://pith.science/pith/SSGON2DJYHUUOLM5NVCB5YFPG5","bundle":"https://pith.science/pith/SSGON2DJYHUUOLM5NVCB5YFPG5/bundle.json","state":"https://pith.science/pith/SSGON2DJYHUUOLM5NVCB5YFPG5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SSGON2DJYHUUOLM5NVCB5YFPG5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:SSGON2DJYHUUOLM5NVCB5YFPG5","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":"0011f26ccf390400bb1b4305986523d9beece881379712564c89deeab85e170e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-09T10:43:11Z","title_canon_sha256":"031763eac56d75acc71d0fd7e0d9074e678afb70f25ee82f9b2681e64f574985"},"schema_version":"1.0","source":{"id":"2507.06730","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06730","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06730v1","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06730","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"pith_short_12","alias_value":"SSGON2DJYHUU","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"pith_short_16","alias_value":"SSGON2DJYHUUOLM5","created_at":"2026-07-05T11:34:22Z"},{"alias_kind":"pith_short_8","alias_value":"SSGON2DJ","created_at":"2026-07-05T11:34:22Z"}],"graph_snapshots":[{"event_id":"sha256:ba84db50867739c77331bda338afb29dd52756ffb476b7bba4b7eb3f97195bc2","target":"graph","created_at":"2026-07-05T11:34:22Z","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/2507.06730/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Sierpi\\'{n}ski product $G \\otimes _f H$ of graphs $G$ and $H$ with respect to a function $f \\colon V(G)\\rightarrow V(H)$ has the vertex set $V(G)\\times V(H)$. For every $g\\in V(G)$ it contains a disjoint copy $gH$ of $H$, and for every edge $gg'$ of $G$ there is the edge $(g,f(g'))(g',f(g))$ between $gH$ and $g'H$. In this paper, the Sierpi\\'{n}ski packing chromatic number is defined as the minimum of $\\chi_{\\rho}(G\\otimes _f H)$ over all functions $f$, where $\\chi_{\\rho}(X)$ is the packing chromatic number of $X$. The upper Sierpi\\'{n}ski packing chromatic number is analogously defined as","authors_text":"P\\v{r}emysl Holub, Sandi Klav\\v{z}ar","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-09T10:43:11Z","title":"On Sierpi\\'{n}ski packing chromatic number and recognition of Sierpi\\'{n}ski products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06730","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":"sha256:1a340874ad45b5d8d2078a4495c21b2da98bf7fb406f73eff2654edbad8126c0","target":"record","created_at":"2026-07-05T11:34:22Z","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":"0011f26ccf390400bb1b4305986523d9beece881379712564c89deeab85e170e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-09T10:43:11Z","title_canon_sha256":"031763eac56d75acc71d0fd7e0d9074e678afb70f25ee82f9b2681e64f574985"},"schema_version":"1.0","source":{"id":"2507.06730","kind":"arxiv","version":1}},"canonical_sha256":"948ce6e869c1e9472d9d6d441ee0af3775d565ccdb8c032833ef6ba8f6221dae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"948ce6e869c1e9472d9d6d441ee0af3775d565ccdb8c032833ef6ba8f6221dae","first_computed_at":"2026-07-05T11:34:22.168667Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:34:22.168667Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1Wc0PcEfP0p8Xaq7an+IFLNSEi7IU8v/S8tLpkfOC0mQ6DmoQ7xTpXbzjOPqfEwprujCfTzRr39RHSMHTHQhBw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:34:22.169083Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.06730","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1a340874ad45b5d8d2078a4495c21b2da98bf7fb406f73eff2654edbad8126c0","sha256:ba84db50867739c77331bda338afb29dd52756ffb476b7bba4b7eb3f97195bc2"],"state_sha256":"15241600d46c34439db9e8759b07dc6917aad5b64ff0510b7caf14be66f1f1e5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a+wgVzD4uLp8q5pOyt0ef0yxFBWJGnZ/Td8pcjq0Ra/w1SpBNnwMN2oXsrC/KQOY46wcEYavu4v5sU4tcSR3Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T11:51:01.312653Z","bundle_sha256":"b01ad34eb8956ec40113c0496d56fbe501a4a873f4f9edf3dd0b836b088b308a"}}