{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:RO3VRN2ZLUXHS5ZI7UV6QB4WQA","short_pith_number":"pith:RO3VRN2Z","canonical_record":{"source":{"id":"1712.04405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-12-12T17:46:08Z","cross_cats_sorted":[],"title_canon_sha256":"14124e53ab26f06796b9022d6f64b7c7a1d061acb6c383345f12df6dac122962","abstract_canon_sha256":"ddb392cdaf4c07f7fe58c219eb16f875e808d4aa955eaf469c2533f44edc4716"},"schema_version":"1.0"},"canonical_sha256":"8bb758b7595d2e797728fd2be8079680323960f24327f9e46f4e84babb509b6d","source":{"kind":"arxiv","id":"1712.04405","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.04405","created_at":"2026-05-17T23:43:09Z"},{"alias_kind":"arxiv_version","alias_value":"1712.04405v1","created_at":"2026-05-17T23:43:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.04405","created_at":"2026-05-17T23:43:09Z"},{"alias_kind":"pith_short_12","alias_value":"RO3VRN2ZLUXH","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"RO3VRN2ZLUXHS5ZI","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"RO3VRN2Z","created_at":"2026-05-18T12:31:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:RO3VRN2ZLUXHS5ZI7UV6QB4WQA","target":"record","payload":{"canonical_record":{"source":{"id":"1712.04405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-12-12T17:46:08Z","cross_cats_sorted":[],"title_canon_sha256":"14124e53ab26f06796b9022d6f64b7c7a1d061acb6c383345f12df6dac122962","abstract_canon_sha256":"ddb392cdaf4c07f7fe58c219eb16f875e808d4aa955eaf469c2533f44edc4716"},"schema_version":"1.0"},"canonical_sha256":"8bb758b7595d2e797728fd2be8079680323960f24327f9e46f4e84babb509b6d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:09.544643Z","signature_b64":"xFaR8l28gNzTu4EUcjXqGpkpoQpKTAoNg7+SKirz5eD6nibVjE/gUyvbnOlynT8BhvG28JTnKwN/nKb7FOJBDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8bb758b7595d2e797728fd2be8079680323960f24327f9e46f4e84babb509b6d","last_reissued_at":"2026-05-17T23:43:09.543988Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:09.543988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.04405","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":"2026-05-17T23:43:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mLnTdTL48shNz2HwAopDHLRyAa0m5+fb/hB59zIibxRyhZwVUgVHOBziq7SyG3f6er8l2M4Ls0gzuaGn2vO7Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T05:12:53.672021Z"},"content_sha256":"87a5fca7a51ff8831677951e8da9410aed8b658f873474fb45e33102920df43d","schema_version":"1.0","event_id":"sha256:87a5fca7a51ff8831677951e8da9410aed8b658f873474fb45e33102920df43d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:RO3VRN2ZLUXHS5ZI7UV6QB4WQA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal height companion matrices for Euclid polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Eunice Y. S. Chan, Robert M. Corless","submitted_at":"2017-12-12T17:46:08Z","abstract_excerpt":"We define Euclid polynomials $E_{k+1}(\\lambda) = E_{k}(\\lambda)\\left(E_{k}(\\lambda) - 1\\right) + 1$ and $E_{1}(\\lambda) = \\lambda + 1$ in analogy to Euclid numbers $e_k = E_{k}(1)$. We show how to construct companion matrices $\\mathbb{E}_k$, so $E_k(\\lambda) = \\operatorname{det}\\left(\\lambda\\mathbf{I} - \\mathbb{E}_{k}\\right)$, of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(\\lambda)$). We prove various properties of these objects, and give experimental confirmation of some unproved properties."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04405","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":""},"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":"2026-05-17T23:43:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5JCYP+vnKiW5K9uSLiCk+bLvDfUV+5W3Isl3cCHykF3bXM3VHs0rL8GXYNTXuDwGqKlDibD+bjxf8gnHTN0sBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T05:12:53.672573Z"},"content_sha256":"b7f27eb45d606df2fcb59ba33a1c67eabc6888e4576932ae78e0c067129f5c86","schema_version":"1.0","event_id":"sha256:b7f27eb45d606df2fcb59ba33a1c67eabc6888e4576932ae78e0c067129f5c86"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RO3VRN2ZLUXHS5ZI7UV6QB4WQA/bundle.json","state_url":"https://pith.science/pith/RO3VRN2ZLUXHS5ZI7UV6QB4WQA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RO3VRN2ZLUXHS5ZI7UV6QB4WQA/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":"2026-06-26T05:12:53Z","links":{"resolver":"https://pith.science/pith/RO3VRN2ZLUXHS5ZI7UV6QB4WQA","bundle":"https://pith.science/pith/RO3VRN2ZLUXHS5ZI7UV6QB4WQA/bundle.json","state":"https://pith.science/pith/RO3VRN2ZLUXHS5ZI7UV6QB4WQA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RO3VRN2ZLUXHS5ZI7UV6QB4WQA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:RO3VRN2ZLUXHS5ZI7UV6QB4WQA","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":"ddb392cdaf4c07f7fe58c219eb16f875e808d4aa955eaf469c2533f44edc4716","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-12-12T17:46:08Z","title_canon_sha256":"14124e53ab26f06796b9022d6f64b7c7a1d061acb6c383345f12df6dac122962"},"schema_version":"1.0","source":{"id":"1712.04405","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.04405","created_at":"2026-05-17T23:43:09Z"},{"alias_kind":"arxiv_version","alias_value":"1712.04405v1","created_at":"2026-05-17T23:43:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.04405","created_at":"2026-05-17T23:43:09Z"},{"alias_kind":"pith_short_12","alias_value":"RO3VRN2ZLUXH","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"RO3VRN2ZLUXHS5ZI","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"RO3VRN2Z","created_at":"2026-05-18T12:31:39Z"}],"graph_snapshots":[{"event_id":"sha256:b7f27eb45d606df2fcb59ba33a1c67eabc6888e4576932ae78e0c067129f5c86","target":"graph","created_at":"2026-05-17T23:43:09Z","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 define Euclid polynomials $E_{k+1}(\\lambda) = E_{k}(\\lambda)\\left(E_{k}(\\lambda) - 1\\right) + 1$ and $E_{1}(\\lambda) = \\lambda + 1$ in analogy to Euclid numbers $e_k = E_{k}(1)$. We show how to construct companion matrices $\\mathbb{E}_k$, so $E_k(\\lambda) = \\operatorname{det}\\left(\\lambda\\mathbf{I} - \\mathbb{E}_{k}\\right)$, of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(\\lambda)$). We prove various properties of these objects, and give experimental confirmation of some unproved properties.","authors_text":"Eunice Y. S. Chan, Robert M. Corless","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-12-12T17:46:08Z","title":"Minimal height companion matrices for Euclid polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04405","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":"sha256:87a5fca7a51ff8831677951e8da9410aed8b658f873474fb45e33102920df43d","target":"record","created_at":"2026-05-17T23:43:09Z","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":"ddb392cdaf4c07f7fe58c219eb16f875e808d4aa955eaf469c2533f44edc4716","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-12-12T17:46:08Z","title_canon_sha256":"14124e53ab26f06796b9022d6f64b7c7a1d061acb6c383345f12df6dac122962"},"schema_version":"1.0","source":{"id":"1712.04405","kind":"arxiv","version":1}},"canonical_sha256":"8bb758b7595d2e797728fd2be8079680323960f24327f9e46f4e84babb509b6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8bb758b7595d2e797728fd2be8079680323960f24327f9e46f4e84babb509b6d","first_computed_at":"2026-05-17T23:43:09.543988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:09.543988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xFaR8l28gNzTu4EUcjXqGpkpoQpKTAoNg7+SKirz5eD6nibVjE/gUyvbnOlynT8BhvG28JTnKwN/nKb7FOJBDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:09.544643Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.04405","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87a5fca7a51ff8831677951e8da9410aed8b658f873474fb45e33102920df43d","sha256:b7f27eb45d606df2fcb59ba33a1c67eabc6888e4576932ae78e0c067129f5c86"],"state_sha256":"52acf3367886cd1eb82c8b6e075e147e2838a15555c930a28f32d8773bbc2191"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LOtCYFZvCf9LLnUu/nmTfYG/WtN0CnYxqgf9ZWy9wK9FT2oePShzUrK+pm92SYpG4bB/uW4sytQ+MxAreHY8DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T05:12:53.675248Z","bundle_sha256":"647bc0f1c7aef57d0e468b3f9030e23f0b23bd6a4374335b7cfdac4ccd2b2c26"}}