{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TFPRZS6OB7OGLOLV22XFL6TV3N","short_pith_number":"pith:TFPRZS6O","canonical_record":{"source":{"id":"2606.01761","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T06:36:41Z","cross_cats_sorted":[],"title_canon_sha256":"0a85f8144a9f15764a57ad0e5897d2b9d4ce5975f6485ad198fc7060ec2a2963","abstract_canon_sha256":"72063bc956d87cd99a39cd6ec6db5ce0ffb9bc113e0e0446c6fe0911b63026ea"},"schema_version":"1.0"},"canonical_sha256":"995f1ccbce0fdc65b975d6ae55fa75db76e7f0518ea20655025d46181ed6ec50","source":{"kind":"arxiv","id":"2606.01761","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.01761","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"arxiv_version","alias_value":"2606.01761v1","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.01761","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"pith_short_12","alias_value":"TFPRZS6OB7OG","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"pith_short_16","alias_value":"TFPRZS6OB7OGLOLV","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"pith_short_8","alias_value":"TFPRZS6O","created_at":"2026-06-02T02:04:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TFPRZS6OB7OGLOLV22XFL6TV3N","target":"record","payload":{"canonical_record":{"source":{"id":"2606.01761","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T06:36:41Z","cross_cats_sorted":[],"title_canon_sha256":"0a85f8144a9f15764a57ad0e5897d2b9d4ce5975f6485ad198fc7060ec2a2963","abstract_canon_sha256":"72063bc956d87cd99a39cd6ec6db5ce0ffb9bc113e0e0446c6fe0911b63026ea"},"schema_version":"1.0"},"canonical_sha256":"995f1ccbce0fdc65b975d6ae55fa75db76e7f0518ea20655025d46181ed6ec50","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:55.956585Z","signature_b64":"zmGUjs7zyFEGUWyEz0giW958dcLo5G0htoUHmmmzA9QB+UjQC+j4QYHZnZCNMDw1NH6+S6pS3W2HwzbheahICg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"995f1ccbce0fdc65b975d6ae55fa75db76e7f0518ea20655025d46181ed6ec50","last_reissued_at":"2026-06-02T02:04:55.956176Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:55.956176Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.01761","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":"2026-06-02T02:04:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O9AVb4ziYH3BOJw2tTtCgPOepW4tg5MSGjQfqjvtPL04zPc+5Ca98XieAnj+Qf6ixdQGleXvvT6d1izHGhXEBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:35:57.386208Z"},"content_sha256":"1ea9cf5e7348425de961161464487785f6b60993ab4b0e73d2655f9c736927fe","schema_version":"1.0","event_id":"sha256:1ea9cf5e7348425de961161464487785f6b60993ab4b0e73d2655f9c736927fe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TFPRZS6OB7OGLOLV22XFL6TV3N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on the Ratio and Inertia Bounds for the $k$-Independence Number","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Ma, Jun Gao, Oleg Pikhurko","submitted_at":"2026-06-01T06:36:41Z","abstract_excerpt":"The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875--2885] proposed extensions of the classical ratio (for regular graphs) and inertia bounds to the independence number of $G^k$ for $k\\ge 2$.\n  Continuing a line of work comparing these two parameters with other known bounds, we show that the $\\vartheta$-function of L. Lov\\'asz and the w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01761","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/2606.01761/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":"2026-06-02T02:04:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gZIJ/KyMGC0srk3UCNSVzmRTfM93jzYAiy0b5YOpbLrU9XXBhQSnvOl1WmVyFFvRuq7e6E4gvzOr8sesDzQeBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:35:57.386610Z"},"content_sha256":"46167248aa505e6f69bb7be468f7c357090116ca4176f67bbf1748770a3ec2ab","schema_version":"1.0","event_id":"sha256:46167248aa505e6f69bb7be468f7c357090116ca4176f67bbf1748770a3ec2ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TFPRZS6OB7OGLOLV22XFL6TV3N/bundle.json","state_url":"https://pith.science/pith/TFPRZS6OB7OGLOLV22XFL6TV3N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TFPRZS6OB7OGLOLV22XFL6TV3N/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":"2026-06-03T03:35:57Z","links":{"resolver":"https://pith.science/pith/TFPRZS6OB7OGLOLV22XFL6TV3N","bundle":"https://pith.science/pith/TFPRZS6OB7OGLOLV22XFL6TV3N/bundle.json","state":"https://pith.science/pith/TFPRZS6OB7OGLOLV22XFL6TV3N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TFPRZS6OB7OGLOLV22XFL6TV3N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TFPRZS6OB7OGLOLV22XFL6TV3N","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":"72063bc956d87cd99a39cd6ec6db5ce0ffb9bc113e0e0446c6fe0911b63026ea","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T06:36:41Z","title_canon_sha256":"0a85f8144a9f15764a57ad0e5897d2b9d4ce5975f6485ad198fc7060ec2a2963"},"schema_version":"1.0","source":{"id":"2606.01761","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.01761","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"arxiv_version","alias_value":"2606.01761v1","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.01761","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"pith_short_12","alias_value":"TFPRZS6OB7OG","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"pith_short_16","alias_value":"TFPRZS6OB7OGLOLV","created_at":"2026-06-02T02:04:55Z"},{"alias_kind":"pith_short_8","alias_value":"TFPRZS6O","created_at":"2026-06-02T02:04:55Z"}],"graph_snapshots":[{"event_id":"sha256:46167248aa505e6f69bb7be468f7c357090116ca4176f67bbf1748770a3ec2ab","target":"graph","created_at":"2026-06-02T02:04:55Z","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/2606.01761/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875--2885] proposed extensions of the classical ratio (for regular graphs) and inertia bounds to the independence number of $G^k$ for $k\\ge 2$.\n  Continuing a line of work comparing these two parameters with other known bounds, we show that the $\\vartheta$-function of L. Lov\\'asz and the w","authors_text":"Jie Ma, Jun Gao, Oleg Pikhurko","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T06:36:41Z","title":"A note on the Ratio and Inertia Bounds for the $k$-Independence Number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01761","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":"sha256:1ea9cf5e7348425de961161464487785f6b60993ab4b0e73d2655f9c736927fe","target":"record","created_at":"2026-06-02T02:04:55Z","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":"72063bc956d87cd99a39cd6ec6db5ce0ffb9bc113e0e0446c6fe0911b63026ea","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T06:36:41Z","title_canon_sha256":"0a85f8144a9f15764a57ad0e5897d2b9d4ce5975f6485ad198fc7060ec2a2963"},"schema_version":"1.0","source":{"id":"2606.01761","kind":"arxiv","version":1}},"canonical_sha256":"995f1ccbce0fdc65b975d6ae55fa75db76e7f0518ea20655025d46181ed6ec50","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"995f1ccbce0fdc65b975d6ae55fa75db76e7f0518ea20655025d46181ed6ec50","first_computed_at":"2026-06-02T02:04:55.956176Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T02:04:55.956176Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zmGUjs7zyFEGUWyEz0giW958dcLo5G0htoUHmmmzA9QB+UjQC+j4QYHZnZCNMDw1NH6+S6pS3W2HwzbheahICg==","signature_status":"signed_v1","signed_at":"2026-06-02T02:04:55.956585Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.01761","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1ea9cf5e7348425de961161464487785f6b60993ab4b0e73d2655f9c736927fe","sha256:46167248aa505e6f69bb7be468f7c357090116ca4176f67bbf1748770a3ec2ab"],"state_sha256":"8d14d6ce33c0bcacb6bf44ecefdc1eb5f05a6d41e3fa60a5b3f96cc0dee4ab16"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oovjyFAH9w3isCBkyBNlhzxYZX8asfdd2229Hjx2wA4/mVdCmMpINAWlAfgoQHhgVIyw8k+qhY5+mmNXU/GpBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T03:35:57.388685Z","bundle_sha256":"e2f4273b58cf1c3774f3d989d5c367ce80b91fb0866581663dc5309d90460c99"}}