{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:F55RKUQJAEBODLSCTUICEEQNIX","short_pith_number":"pith:F55RKUQJ","canonical_record":{"source":{"id":"1007.2822","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-16T17:59:50Z","cross_cats_sorted":[],"title_canon_sha256":"ff8550d8e850ab258d952d461f51ecff103ac79c23d9288fea51e7077151cd23","abstract_canon_sha256":"d46bd42d530a22a6f63b1a76412ceeaf09b0c72d65af4cebfc407f71fc068854"},"schema_version":"1.0"},"canonical_sha256":"2f7b1552090102e1ae429d1022120d45e31c5843f3cdc58ec5a69d4c910f26b6","source":{"kind":"arxiv","id":"1007.2822","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2822","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2822v2","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2822","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"F55RKUQJAEBO","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"F55RKUQJAEBODLSC","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"F55RKUQJ","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:F55RKUQJAEBODLSCTUICEEQNIX","target":"record","payload":{"canonical_record":{"source":{"id":"1007.2822","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-16T17:59:50Z","cross_cats_sorted":[],"title_canon_sha256":"ff8550d8e850ab258d952d461f51ecff103ac79c23d9288fea51e7077151cd23","abstract_canon_sha256":"d46bd42d530a22a6f63b1a76412ceeaf09b0c72d65af4cebfc407f71fc068854"},"schema_version":"1.0"},"canonical_sha256":"2f7b1552090102e1ae429d1022120d45e31c5843f3cdc58ec5a69d4c910f26b6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:35.965459Z","signature_b64":"2M/eZkCTMkrM6r7nOMHhcrjfzacdYbIoINZ3A5fQEvrmXwljhqfvhJUudDuhkm8cfehFGGGeYDQ3Ya/pPnRdBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f7b1552090102e1ae429d1022120d45e31c5843f3cdc58ec5a69d4c910f26b6","last_reissued_at":"2026-05-18T03:05:35.964789Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:35.964789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.2822","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:05:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xn5DQ8FMU1kV+l8sn2CTrwtooOT5ttfbp68MN3hZ1+MFR1dP234t4xRXFYGHhXJyuOQqUQqRN81l5iMSGdz0DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:46:59.213836Z"},"content_sha256":"22c34bda27b9dd25bcba5d73a5203f89e794ad88a07bc9ec08f859c780654e6f","schema_version":"1.0","event_id":"sha256:22c34bda27b9dd25bcba5d73a5203f89e794ad88a07bc9ec08f859c780654e6f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:F55RKUQJAEBODLSCTUICEEQNIX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal decomposition of binary forms with respect to tangential projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alessandra Bernardi, Edoardo Ballico","submitted_at":"2010-07-16T17:59:50Z","abstract_excerpt":"Let $C\\subset \\mathbb{P}^n$ be a rational normal curve and let $\\ell_O:\\mathbb{P}^{n+1}\\dashrightarrow \\mathbb{P}^n$ be any tangential projection form a point $O\\in T_AC$ where $A\\in C$. Hence $X:= \\ell_O(C)\\subset \\mathbb{P}^n$ is a linearly normal cuspidal curve with degree $n+1$. For any $P = \\ell_O(B)$, $B\\in \\mathbb{P}^{n+1}$, the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\\subset X$ whose linear span contains $P$. Here we describe $r_X(P)$ in terms of the schemes computing the $C$-rank or the border $C$-rank of $B$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2822","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T03:05:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5ijZztB8SIX7LvHAo3LaCxG2p99QYjMx3JcgQDRKpRRLrAkMgb6aJP/3wDkSnStIEuIKwG8CDhucZFRual/BBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:46:59.214204Z"},"content_sha256":"77ac172ca1c8ca91913f3d255efbb284d77737ae129172b69dee48feb5a06977","schema_version":"1.0","event_id":"sha256:77ac172ca1c8ca91913f3d255efbb284d77737ae129172b69dee48feb5a06977"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/bundle.json","state_url":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F55RKUQJAEBODLSCTUICEEQNIX/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":"2026-06-01T06:46:59Z","links":{"resolver":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX","bundle":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/bundle.json","state":"https://pith.science/pith/F55RKUQJAEBODLSCTUICEEQNIX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F55RKUQJAEBODLSCTUICEEQNIX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:F55RKUQJAEBODLSCTUICEEQNIX","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":"d46bd42d530a22a6f63b1a76412ceeaf09b0c72d65af4cebfc407f71fc068854","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-16T17:59:50Z","title_canon_sha256":"ff8550d8e850ab258d952d461f51ecff103ac79c23d9288fea51e7077151cd23"},"schema_version":"1.0","source":{"id":"1007.2822","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2822","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2822v2","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2822","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"F55RKUQJAEBO","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"F55RKUQJAEBODLSC","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"F55RKUQJ","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:77ac172ca1c8ca91913f3d255efbb284d77737ae129172b69dee48feb5a06977","target":"graph","created_at":"2026-05-18T03:05:35Z","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"},"paper":{"abstract_excerpt":"Let $C\\subset \\mathbb{P}^n$ be a rational normal curve and let $\\ell_O:\\mathbb{P}^{n+1}\\dashrightarrow \\mathbb{P}^n$ be any tangential projection form a point $O\\in T_AC$ where $A\\in C$. Hence $X:= \\ell_O(C)\\subset \\mathbb{P}^n$ is a linearly normal cuspidal curve with degree $n+1$. For any $P = \\ell_O(B)$, $B\\in \\mathbb{P}^{n+1}$, the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\\subset X$ whose linear span contains $P$. Here we describe $r_X(P)$ in terms of the schemes computing the $C$-rank or the border $C$-rank of $B$.","authors_text":"Alessandra Bernardi, Edoardo Ballico","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-16T17:59:50Z","title":"Minimal decomposition of binary forms with respect to tangential projections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2822","kind":"arxiv","version":2},"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":"sha256:22c34bda27b9dd25bcba5d73a5203f89e794ad88a07bc9ec08f859c780654e6f","target":"record","created_at":"2026-05-18T03:05:35Z","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":"d46bd42d530a22a6f63b1a76412ceeaf09b0c72d65af4cebfc407f71fc068854","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-07-16T17:59:50Z","title_canon_sha256":"ff8550d8e850ab258d952d461f51ecff103ac79c23d9288fea51e7077151cd23"},"schema_version":"1.0","source":{"id":"1007.2822","kind":"arxiv","version":2}},"canonical_sha256":"2f7b1552090102e1ae429d1022120d45e31c5843f3cdc58ec5a69d4c910f26b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f7b1552090102e1ae429d1022120d45e31c5843f3cdc58ec5a69d4c910f26b6","first_computed_at":"2026-05-18T03:05:35.964789Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:35.964789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2M/eZkCTMkrM6r7nOMHhcrjfzacdYbIoINZ3A5fQEvrmXwljhqfvhJUudDuhkm8cfehFGGGeYDQ3Ya/pPnRdBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:35.965459Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.2822","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22c34bda27b9dd25bcba5d73a5203f89e794ad88a07bc9ec08f859c780654e6f","sha256:77ac172ca1c8ca91913f3d255efbb284d77737ae129172b69dee48feb5a06977"],"state_sha256":"e71e2d3f6a0c45dab7d0e42136c9a21821f3fd94863aebf2816869f3a95064c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vIThvBnrRFKj/AJUrVVjuOjO87k5WidvmyTjxqsSWV3HHUn1yjrPGuQUgndfgLGi2zuVJOmMSJmYlrJudR13Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T06:46:59.216362Z","bundle_sha256":"c194ae4937cbe5dd580b9ad4810008da7a846dde9c123101568de04536856d4b"}}