{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:H33HSU67VR4PQRBUIXO432LIQW","short_pith_number":"pith:H33HSU67","canonical_record":{"source":{"id":"1303.6915","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-03-27T18:04:38Z","cross_cats_sorted":[],"title_canon_sha256":"2b40354cf171bb302c4c74bdb71c0a5964c8881ed567316462a4e8f1a0f855bf","abstract_canon_sha256":"7f6b767bac0d8a4415c702889a34c145f0164842d9282610ff669257b17996d9"},"schema_version":"1.0"},"canonical_sha256":"3ef67953dfac78f8443445ddcde96885a82b170347966403f14cd3820cf13ec1","source":{"kind":"arxiv","id":"1303.6915","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.6915","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"arxiv_version","alias_value":"1303.6915v3","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.6915","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"pith_short_12","alias_value":"H33HSU67VR4P","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"H33HSU67VR4PQRBU","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"H33HSU67","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:H33HSU67VR4PQRBUIXO432LIQW","target":"record","payload":{"canonical_record":{"source":{"id":"1303.6915","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-03-27T18:04:38Z","cross_cats_sorted":[],"title_canon_sha256":"2b40354cf171bb302c4c74bdb71c0a5964c8881ed567316462a4e8f1a0f855bf","abstract_canon_sha256":"7f6b767bac0d8a4415c702889a34c145f0164842d9282610ff669257b17996d9"},"schema_version":"1.0"},"canonical_sha256":"3ef67953dfac78f8443445ddcde96885a82b170347966403f14cd3820cf13ec1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:58.278178Z","signature_b64":"FqOiwGVKljiLKqKC82YdxzbcnglNWwxGGoIhpwdGMVbcZeCl4Ost0OVgvpyJPKLCKToRpm4tLBz8LKjE7zvgDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ef67953dfac78f8443445ddcde96885a82b170347966403f14cd3820cf13ec1","last_reissued_at":"2026-05-18T03:25:58.277545Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:58.277545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.6915","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":"2026-05-18T03:25:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7Xvaq3g9dJtnAV4faSGCNxB9KR+03ZknPDcvRD8htj6+Z0jAj6a25xBcN9RqJ0+cDVB6lPw9NRBLxg+FCKV+Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T00:29:52.734889Z"},"content_sha256":"49404dac9f1c9fb983beb4baffeb8c99aad3173823c9917cdc0871247968c25d","schema_version":"1.0","event_id":"sha256:49404dac9f1c9fb983beb4baffeb8c99aad3173823c9917cdc0871247968c25d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:H33HSU67VR4PQRBUIXO432LIQW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Refined methods for the identifiability of tensors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Cristiano Bocci, Giorgio Ottaviani, Luca Chiantini","submitted_at":"2013-03-27T18:04:38Z","abstract_excerpt":"We prove that the general tensor of size 2^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.9997 (2^n)/(n+1) (the constant 1 being the optimal value). Similarly, the general tensor of size 3^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.998 (3^n)/(2n+1) (the constant 1 being the optimal value).\n  Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6915","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":""},"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":"2026-05-18T03:25:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lVTJhk5EORsIuF9GXBvvUEj66L4fQAWA4v3EbKn3H/7tNcnLEKproAA2eUNVlqAyj/XjHNGbbMjRZ2kKxHQfDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T00:29:52.735480Z"},"content_sha256":"e5c9860c34030f39528da80679201cae294fdc5c2af05c04c9fea831f3996354","schema_version":"1.0","event_id":"sha256:e5c9860c34030f39528da80679201cae294fdc5c2af05c04c9fea831f3996354"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H33HSU67VR4PQRBUIXO432LIQW/bundle.json","state_url":"https://pith.science/pith/H33HSU67VR4PQRBUIXO432LIQW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H33HSU67VR4PQRBUIXO432LIQW/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":"2026-05-30T00:29:52Z","links":{"resolver":"https://pith.science/pith/H33HSU67VR4PQRBUIXO432LIQW","bundle":"https://pith.science/pith/H33HSU67VR4PQRBUIXO432LIQW/bundle.json","state":"https://pith.science/pith/H33HSU67VR4PQRBUIXO432LIQW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H33HSU67VR4PQRBUIXO432LIQW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:H33HSU67VR4PQRBUIXO432LIQW","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":"7f6b767bac0d8a4415c702889a34c145f0164842d9282610ff669257b17996d9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-03-27T18:04:38Z","title_canon_sha256":"2b40354cf171bb302c4c74bdb71c0a5964c8881ed567316462a4e8f1a0f855bf"},"schema_version":"1.0","source":{"id":"1303.6915","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.6915","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"arxiv_version","alias_value":"1303.6915v3","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.6915","created_at":"2026-05-18T03:25:58Z"},{"alias_kind":"pith_short_12","alias_value":"H33HSU67VR4P","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"H33HSU67VR4PQRBU","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"H33HSU67","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:e5c9860c34030f39528da80679201cae294fdc5c2af05c04c9fea831f3996354","target":"graph","created_at":"2026-05-18T03:25:58Z","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":"We prove that the general tensor of size 2^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.9997 (2^n)/(n+1) (the constant 1 being the optimal value). Similarly, the general tensor of size 3^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.998 (3^n)/(2n+1) (the constant 1 being the optimal value).\n  Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker.","authors_text":"Cristiano Bocci, Giorgio Ottaviani, Luca Chiantini","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-03-27T18:04:38Z","title":"Refined methods for the identifiability of tensors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.6915","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":"sha256:49404dac9f1c9fb983beb4baffeb8c99aad3173823c9917cdc0871247968c25d","target":"record","created_at":"2026-05-18T03:25:58Z","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":"7f6b767bac0d8a4415c702889a34c145f0164842d9282610ff669257b17996d9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-03-27T18:04:38Z","title_canon_sha256":"2b40354cf171bb302c4c74bdb71c0a5964c8881ed567316462a4e8f1a0f855bf"},"schema_version":"1.0","source":{"id":"1303.6915","kind":"arxiv","version":3}},"canonical_sha256":"3ef67953dfac78f8443445ddcde96885a82b170347966403f14cd3820cf13ec1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3ef67953dfac78f8443445ddcde96885a82b170347966403f14cd3820cf13ec1","first_computed_at":"2026-05-18T03:25:58.277545Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:58.277545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FqOiwGVKljiLKqKC82YdxzbcnglNWwxGGoIhpwdGMVbcZeCl4Ost0OVgvpyJPKLCKToRpm4tLBz8LKjE7zvgDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:58.278178Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.6915","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49404dac9f1c9fb983beb4baffeb8c99aad3173823c9917cdc0871247968c25d","sha256:e5c9860c34030f39528da80679201cae294fdc5c2af05c04c9fea831f3996354"],"state_sha256":"398c585b5681ba73d04e1d78845ac340d8aa4106a87aee4567926fe0f7e97802"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3g3HGEL5bRKQi3/syTkKj3F8MbK7JUG/79zEQp+i69v/zR8ZZpEoEe99egcfZ3c0k+TIrXvzCDYlTW/ISBHAAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T00:29:52.738944Z","bundle_sha256":"21466567140d88112556cbbf1d4bbd86593c772b4771667efde8f97a3ab006f6"}}