{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:5PBU3UBG7ACUVBAJ3RZUT6WABI","short_pith_number":"pith:5PBU3UBG","canonical_record":{"source":{"id":"2106.05650","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2021-06-10T10:45:38Z","cross_cats_sorted":[],"title_canon_sha256":"d0742db4123f60b124f19cd028fb9ee8b2a835626e01b3749060b302a428d2b4","abstract_canon_sha256":"9f45a3dea6b82d289d22588ea3a6a8a099e907813bb1369090ebdcdf77055927"},"schema_version":"1.0"},"canonical_sha256":"ebc34dd026f8054a8409dc7349fac00a06d5a9d4a5a8b1b2fe3b7f69b7bc2a12","source":{"kind":"arxiv","id":"2106.05650","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.05650","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"arxiv_version","alias_value":"2106.05650v2","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.05650","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"pith_short_12","alias_value":"5PBU3UBG7ACU","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"pith_short_16","alias_value":"5PBU3UBG7ACUVBAJ","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"pith_short_8","alias_value":"5PBU3UBG","created_at":"2026-07-05T03:03:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:5PBU3UBG7ACUVBAJ3RZUT6WABI","target":"record","payload":{"canonical_record":{"source":{"id":"2106.05650","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2021-06-10T10:45:38Z","cross_cats_sorted":[],"title_canon_sha256":"d0742db4123f60b124f19cd028fb9ee8b2a835626e01b3749060b302a428d2b4","abstract_canon_sha256":"9f45a3dea6b82d289d22588ea3a6a8a099e907813bb1369090ebdcdf77055927"},"schema_version":"1.0"},"canonical_sha256":"ebc34dd026f8054a8409dc7349fac00a06d5a9d4a5a8b1b2fe3b7f69b7bc2a12","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:03:08.717331Z","signature_b64":"+FQgbOGe4pzfQytEOREUjOt3klllDRrCKohly5gynSFdoRQnq4SXzCD9XJ59oFacLz7Oik8UwaR1DaOL2pE2Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebc34dd026f8054a8409dc7349fac00a06d5a9d4a5a8b1b2fe3b7f69b7bc2a12","last_reissued_at":"2026-07-05T03:03:08.716918Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:03:08.716918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2106.05650","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":"2026-07-05T03:03:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Omwp8z0EYH3aDM5fQvslPpWmLkr/4oerWwekxs2ez4zJF/FQa58/e1RNJB/RPtWBCfIx2x8TrWzQ/jpggl4YCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T14:49:31.929048Z"},"content_sha256":"c28bc5e3d8f1c22fc3e5d334b023d2b108a0e70402960c7f5e3444b05a99f075","schema_version":"1.0","event_id":"sha256:c28bc5e3d8f1c22fc3e5d334b023d2b108a0e70402960c7f5e3444b05a99f075"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:5PBU3UBG7ACUVBAJ3RZUT6WABI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Scaled Relative Graph of a Linear Operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Richard Pates","submitted_at":"2021-06-10T10:45:38Z","abstract_excerpt":"The scaled relative graph (SRG) of an operator is a subset of the complex plane. It captures several salient features of an operator, such as contractiveness, and can be used to reveal the geometric nature of many of the inequality based arguments used in the convergence analyses of fixed point iterations. In this paper we show that the SRG of a linear operator can be determined from the numerical range of a closely related linear operator. Furthermore we demonstrate that the SRG of a linear operator has a range of spectral and convexity properties, and satisfies an analogue of Hildebrant's th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.05650","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.05650/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":"2026-07-05T03:03:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9v93cOWBcNjmn47Bqecqdu45NEyWk+mXtuMdZS2TaXkxplRjRoy1N3siYB9ZvkpXli6bmf9rIyCy8FyVlnJdAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T14:49:31.929982Z"},"content_sha256":"d58d76fd275db64d0bea6dd24f8e3324abe09dea1058541fc52cf357c4de2e49","schema_version":"1.0","event_id":"sha256:d58d76fd275db64d0bea6dd24f8e3324abe09dea1058541fc52cf357c4de2e49"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5PBU3UBG7ACUVBAJ3RZUT6WABI/bundle.json","state_url":"https://pith.science/pith/5PBU3UBG7ACUVBAJ3RZUT6WABI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5PBU3UBG7ACUVBAJ3RZUT6WABI/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":"2026-08-13T14:49:31Z","links":{"resolver":"https://pith.science/pith/5PBU3UBG7ACUVBAJ3RZUT6WABI","bundle":"https://pith.science/pith/5PBU3UBG7ACUVBAJ3RZUT6WABI/bundle.json","state":"https://pith.science/pith/5PBU3UBG7ACUVBAJ3RZUT6WABI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5PBU3UBG7ACUVBAJ3RZUT6WABI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:5PBU3UBG7ACUVBAJ3RZUT6WABI","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":"9f45a3dea6b82d289d22588ea3a6a8a099e907813bb1369090ebdcdf77055927","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2021-06-10T10:45:38Z","title_canon_sha256":"d0742db4123f60b124f19cd028fb9ee8b2a835626e01b3749060b302a428d2b4"},"schema_version":"1.0","source":{"id":"2106.05650","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.05650","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"arxiv_version","alias_value":"2106.05650v2","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.05650","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"pith_short_12","alias_value":"5PBU3UBG7ACU","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"pith_short_16","alias_value":"5PBU3UBG7ACUVBAJ","created_at":"2026-07-05T03:03:08Z"},{"alias_kind":"pith_short_8","alias_value":"5PBU3UBG","created_at":"2026-07-05T03:03:08Z"}],"graph_snapshots":[{"event_id":"sha256:d58d76fd275db64d0bea6dd24f8e3324abe09dea1058541fc52cf357c4de2e49","target":"graph","created_at":"2026-07-05T03:03:08Z","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/2106.05650/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The scaled relative graph (SRG) of an operator is a subset of the complex plane. It captures several salient features of an operator, such as contractiveness, and can be used to reveal the geometric nature of many of the inequality based arguments used in the convergence analyses of fixed point iterations. In this paper we show that the SRG of a linear operator can be determined from the numerical range of a closely related linear operator. Furthermore we demonstrate that the SRG of a linear operator has a range of spectral and convexity properties, and satisfies an analogue of Hildebrant's th","authors_text":"Richard Pates","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2021-06-10T10:45:38Z","title":"The Scaled Relative Graph of a Linear Operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.05650","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:c28bc5e3d8f1c22fc3e5d334b023d2b108a0e70402960c7f5e3444b05a99f075","target":"record","created_at":"2026-07-05T03:03:08Z","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":"9f45a3dea6b82d289d22588ea3a6a8a099e907813bb1369090ebdcdf77055927","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2021-06-10T10:45:38Z","title_canon_sha256":"d0742db4123f60b124f19cd028fb9ee8b2a835626e01b3749060b302a428d2b4"},"schema_version":"1.0","source":{"id":"2106.05650","kind":"arxiv","version":2}},"canonical_sha256":"ebc34dd026f8054a8409dc7349fac00a06d5a9d4a5a8b1b2fe3b7f69b7bc2a12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ebc34dd026f8054a8409dc7349fac00a06d5a9d4a5a8b1b2fe3b7f69b7bc2a12","first_computed_at":"2026-07-05T03:03:08.716918Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:03:08.716918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+FQgbOGe4pzfQytEOREUjOt3klllDRrCKohly5gynSFdoRQnq4SXzCD9XJ59oFacLz7Oik8UwaR1DaOL2pE2Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:03:08.717331Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.05650","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c28bc5e3d8f1c22fc3e5d334b023d2b108a0e70402960c7f5e3444b05a99f075","sha256:d58d76fd275db64d0bea6dd24f8e3324abe09dea1058541fc52cf357c4de2e49"],"state_sha256":"4c5833fcda8182179cfe4bf648e2e307ab65ac0bfb0141027cf0fa974f78d69f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KlMNLcSDXkNvwlVqhThqrx7pybCE5mFtlmTpvLndlQP0HRlaynh+SajO4OaIZf3Yj/X8hNBXajwPqmT/q0wODQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T14:49:31.952294Z","bundle_sha256":"e6f5e2b389f3070e5461ea641e81d805981052872a925a6f11ac5b5c84035bd1"}}