{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:L6GHDY55CMYA5M76B2S27JNZJF","short_pith_number":"pith:L6GHDY55","canonical_record":{"source":{"id":"2407.14048","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-07-19T06:03:26Z","cross_cats_sorted":["math.CT","math.OA"],"title_canon_sha256":"e893aa00b7be1732012d4afd2babb22c67ce30a7e8b7d3ca57174553114acae4","abstract_canon_sha256":"9081581c8634fd2d5b2905603cfeae5fca2d7234a6860b4a91d02691f2d8f760"},"schema_version":"1.0"},"canonical_sha256":"5f8c71e3bd13300eb3fe0ea5afa5b9496933c537d77ddb3bb2912ee26067a5a6","source":{"kind":"arxiv","id":"2407.14048","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.14048","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"arxiv_version","alias_value":"2407.14048v3","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.14048","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"pith_short_12","alias_value":"L6GHDY55CMYA","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"pith_short_16","alias_value":"L6GHDY55CMYA5M76","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"pith_short_8","alias_value":"L6GHDY55","created_at":"2026-06-02T02:04:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:L6GHDY55CMYA5M76B2S27JNZJF","target":"record","payload":{"canonical_record":{"source":{"id":"2407.14048","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-07-19T06:03:26Z","cross_cats_sorted":["math.CT","math.OA"],"title_canon_sha256":"e893aa00b7be1732012d4afd2babb22c67ce30a7e8b7d3ca57174553114acae4","abstract_canon_sha256":"9081581c8634fd2d5b2905603cfeae5fca2d7234a6860b4a91d02691f2d8f760"},"schema_version":"1.0"},"canonical_sha256":"5f8c71e3bd13300eb3fe0ea5afa5b9496933c537d77ddb3bb2912ee26067a5a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:04.310648Z","signature_b64":"LKBPAEP/Ss3VQN28HEmpQmy3tOXRIiY+i1k4Lbe/vbiGSsQ8gEb9qtNzeGAT89bRn2n7csrCUl0gV2ZfppfaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f8c71e3bd13300eb3fe0ea5afa5b9496933c537d77ddb3bb2912ee26067a5a6","last_reissued_at":"2026-06-02T02:04:04.310217Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:04.310217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.14048","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T02:04:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+Ln6s8jAYdU6ylNYb283rJUZqPhMGgFW/1Mn8DqSgLnWuwvEE0LqT7jMIEDg6c6D2ztgJksXNEZ3Z6zrsba/BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:59:14.335005Z"},"content_sha256":"6bad9b520f87eb65372f3f2a1b4485fbc795c425dcbfe92c2bcc63f89833a692","schema_version":"1.0","event_id":"sha256:6bad9b520f87eb65372f3f2a1b4485fbc795c425dcbfe92c2bcc63f89833a692"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:L6GHDY55CMYA5M76B2S27JNZJF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Higher-rank trees arising from polyhedral graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT","math.OA"],"primary_cat":"math.CO","authors_text":"David Pask","submitted_at":"2024-07-19T06:03:26Z","abstract_excerpt":"We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek \\cite{Lambek} and Johnstone \\cite{Johnstone} used for monoid and category emedding results. We show that they are planar $k$-trees for $2 \\le k \\le 4$. We also show that higher-rank trees differ from $1$-trees by giving examples of higher-rank trees having properties which are impossible for $1$-trees. Finally, we collect more examples of higher-rank planar trees which are not in our family."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.14048","kind":"arxiv","version":3},"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/2407.14048/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":"2026-06-02T02:04:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NfVto684SLHiGfqFtysyMuT0OEzS9NIHYqYyMvK2xtQUPv8/wL1cu2PSSv2u+LEm7YxZCNRTahWKhpzrBfk+Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T00:59:14.335808Z"},"content_sha256":"665badd6624202b0a3fb7a935b1b0f8b1451a7041f73fdfca00ebc90b7a92cd8","schema_version":"1.0","event_id":"sha256:665badd6624202b0a3fb7a935b1b0f8b1451a7041f73fdfca00ebc90b7a92cd8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L6GHDY55CMYA5M76B2S27JNZJF/bundle.json","state_url":"https://pith.science/pith/L6GHDY55CMYA5M76B2S27JNZJF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L6GHDY55CMYA5M76B2S27JNZJF/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":"2026-06-12T00:59:14Z","links":{"resolver":"https://pith.science/pith/L6GHDY55CMYA5M76B2S27JNZJF","bundle":"https://pith.science/pith/L6GHDY55CMYA5M76B2S27JNZJF/bundle.json","state":"https://pith.science/pith/L6GHDY55CMYA5M76B2S27JNZJF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L6GHDY55CMYA5M76B2S27JNZJF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:L6GHDY55CMYA5M76B2S27JNZJF","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":"9081581c8634fd2d5b2905603cfeae5fca2d7234a6860b4a91d02691f2d8f760","cross_cats_sorted":["math.CT","math.OA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-07-19T06:03:26Z","title_canon_sha256":"e893aa00b7be1732012d4afd2babb22c67ce30a7e8b7d3ca57174553114acae4"},"schema_version":"1.0","source":{"id":"2407.14048","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.14048","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"arxiv_version","alias_value":"2407.14048v3","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.14048","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"pith_short_12","alias_value":"L6GHDY55CMYA","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"pith_short_16","alias_value":"L6GHDY55CMYA5M76","created_at":"2026-06-02T02:04:04Z"},{"alias_kind":"pith_short_8","alias_value":"L6GHDY55","created_at":"2026-06-02T02:04:04Z"}],"graph_snapshots":[{"event_id":"sha256:665badd6624202b0a3fb7a935b1b0f8b1451a7041f73fdfca00ebc90b7a92cd8","target":"graph","created_at":"2026-06-02T02:04:04Z","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/2407.14048/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek \\cite{Lambek} and Johnstone \\cite{Johnstone} used for monoid and category emedding results. We show that they are planar $k$-trees for $2 \\le k \\le 4$. We also show that higher-rank trees differ from $1$-trees by giving examples of higher-rank trees having properties which are impossible for $1$-trees. Finally, we collect more examples of higher-rank planar trees which are not in our family.","authors_text":"David Pask","cross_cats":["math.CT","math.OA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-07-19T06:03:26Z","title":"Higher-rank trees arising from polyhedral graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.14048","kind":"arxiv","version":3},"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":"sha256:6bad9b520f87eb65372f3f2a1b4485fbc795c425dcbfe92c2bcc63f89833a692","target":"record","created_at":"2026-06-02T02:04:04Z","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":"9081581c8634fd2d5b2905603cfeae5fca2d7234a6860b4a91d02691f2d8f760","cross_cats_sorted":["math.CT","math.OA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-07-19T06:03:26Z","title_canon_sha256":"e893aa00b7be1732012d4afd2babb22c67ce30a7e8b7d3ca57174553114acae4"},"schema_version":"1.0","source":{"id":"2407.14048","kind":"arxiv","version":3}},"canonical_sha256":"5f8c71e3bd13300eb3fe0ea5afa5b9496933c537d77ddb3bb2912ee26067a5a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f8c71e3bd13300eb3fe0ea5afa5b9496933c537d77ddb3bb2912ee26067a5a6","first_computed_at":"2026-06-02T02:04:04.310217Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T02:04:04.310217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LKBPAEP/Ss3VQN28HEmpQmy3tOXRIiY+i1k4Lbe/vbiGSsQ8gEb9qtNzeGAT89bRn2n7csrCUl0gV2ZfppfaAw==","signature_status":"signed_v1","signed_at":"2026-06-02T02:04:04.310648Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.14048","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6bad9b520f87eb65372f3f2a1b4485fbc795c425dcbfe92c2bcc63f89833a692","sha256:665badd6624202b0a3fb7a935b1b0f8b1451a7041f73fdfca00ebc90b7a92cd8"],"state_sha256":"6c92e26d12381085a00480ee812e136a95198412870e3a4397f5bfe1b91464ac"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Efb43N/1850d0DSGSULnAw5dxbTRKgPjcZIbrjrP7tsmtR9VKsJHBSm35eKqObkQmVVwAu3iJ01eeUWED8UFCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T00:59:14.340759Z","bundle_sha256":"f4fbaef62ef8a5899edd4b88f83474fffbac4118f03801bb00add0c10ec7da51"}}