{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6PVTVKNJZEVSF4M2SMDWAXEWWY","short_pith_number":"pith:6PVTVKNJ","canonical_record":{"source":{"id":"1510.08109","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-10-27T22:04:36Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"bf47e729dcf67d350407f112429c4468b329113a65076413a94732bc71eafe94","abstract_canon_sha256":"ad00be940917d7c82e97fff5a93486af04fc552dd2006c8cd2fcfd5888ac6f03"},"schema_version":"1.0"},"canonical_sha256":"f3eb3aa9a9c92b22f19a9307605c96b63f0d48dd13ec09baeca1984ec2c06d50","source":{"kind":"arxiv","id":"1510.08109","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.08109","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"arxiv_version","alias_value":"1510.08109v2","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08109","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"pith_short_12","alias_value":"6PVTVKNJZEVS","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6PVTVKNJZEVSF4M2","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6PVTVKNJ","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6PVTVKNJZEVSF4M2SMDWAXEWWY","target":"record","payload":{"canonical_record":{"source":{"id":"1510.08109","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-10-27T22:04:36Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"bf47e729dcf67d350407f112429c4468b329113a65076413a94732bc71eafe94","abstract_canon_sha256":"ad00be940917d7c82e97fff5a93486af04fc552dd2006c8cd2fcfd5888ac6f03"},"schema_version":"1.0"},"canonical_sha256":"f3eb3aa9a9c92b22f19a9307605c96b63f0d48dd13ec09baeca1984ec2c06d50","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:09.268401Z","signature_b64":"XWWlpU2qApo/DTfg95uOKtyNZAIf31GrGZl6iH75+BaV9wYLOApnYha9t/bvY9us8qKXAei99JcX3TDxxWvaDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3eb3aa9a9c92b22f19a9307605c96b63f0d48dd13ec09baeca1984ec2c06d50","last_reissued_at":"2026-05-18T01:26:09.267926Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:09.267926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.08109","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":"2026-05-18T01:26:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wFY54E1AZQXjbTC+XhYzMoj18RzRe2Vgp9bfqX0x0V9IwvIJzcW+AHhJdEWjo4dOYMnXT9wvbj1Fsi3KqaZcCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T07:27:34.785130Z"},"content_sha256":"a5d34aac504c244d740957f4ab63e6f7df2942b0f410f1125ffdec75371195ca","schema_version":"1.0","event_id":"sha256:a5d34aac504c244d740957f4ab63e6f7df2942b0f410f1125ffdec75371195ca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6PVTVKNJZEVSF4M2SMDWAXEWWY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-commutativity of exponential spectrum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.FA","authors_text":"Hubert Klaja, Thomas Ransford","submitted_at":"2015-10-27T22:04:36Z","abstract_excerpt":"In a Banach algebra, the spectrum satisfies $\\sigma(ab)\\setminus\\{0\\} = \\sigma(ba)\\setminus\\{0\\}$ for each pair of elements $a,b$. We show that this is no longer true for the exponential spectrum, thereby solving a problem open since 1992. Our proof depends on the fact that the homotopy group $\\pi_4(GL_2({\\mathbb C}))$ is non-trivial."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08109","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":"2026-05-18T01:26:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vqVRUlGURCf8Qzqv/uSk6lD3qtEmPv8vBT1YMDBgzBGGu684fpdgHhxDO/r7ox6Glg/+3gR8Wyc1IEFZr7oxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T07:27:34.785809Z"},"content_sha256":"e96a394eb819aa29dc6fbe079d01c3aade17cb6a919ec14e98c24688d409a133","schema_version":"1.0","event_id":"sha256:e96a394eb819aa29dc6fbe079d01c3aade17cb6a919ec14e98c24688d409a133"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6PVTVKNJZEVSF4M2SMDWAXEWWY/bundle.json","state_url":"https://pith.science/pith/6PVTVKNJZEVSF4M2SMDWAXEWWY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6PVTVKNJZEVSF4M2SMDWAXEWWY/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":"2026-05-27T07:27:34Z","links":{"resolver":"https://pith.science/pith/6PVTVKNJZEVSF4M2SMDWAXEWWY","bundle":"https://pith.science/pith/6PVTVKNJZEVSF4M2SMDWAXEWWY/bundle.json","state":"https://pith.science/pith/6PVTVKNJZEVSF4M2SMDWAXEWWY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6PVTVKNJZEVSF4M2SMDWAXEWWY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6PVTVKNJZEVSF4M2SMDWAXEWWY","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":"ad00be940917d7c82e97fff5a93486af04fc552dd2006c8cd2fcfd5888ac6f03","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-10-27T22:04:36Z","title_canon_sha256":"bf47e729dcf67d350407f112429c4468b329113a65076413a94732bc71eafe94"},"schema_version":"1.0","source":{"id":"1510.08109","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.08109","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"arxiv_version","alias_value":"1510.08109v2","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08109","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"pith_short_12","alias_value":"6PVTVKNJZEVS","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6PVTVKNJZEVSF4M2","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6PVTVKNJ","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:e96a394eb819aa29dc6fbe079d01c3aade17cb6a919ec14e98c24688d409a133","target":"graph","created_at":"2026-05-18T01:26:09Z","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":"In a Banach algebra, the spectrum satisfies $\\sigma(ab)\\setminus\\{0\\} = \\sigma(ba)\\setminus\\{0\\}$ for each pair of elements $a,b$. We show that this is no longer true for the exponential spectrum, thereby solving a problem open since 1992. Our proof depends on the fact that the homotopy group $\\pi_4(GL_2({\\mathbb C}))$ is non-trivial.","authors_text":"Hubert Klaja, Thomas Ransford","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-10-27T22:04:36Z","title":"Non-commutativity of exponential spectrum"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08109","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:a5d34aac504c244d740957f4ab63e6f7df2942b0f410f1125ffdec75371195ca","target":"record","created_at":"2026-05-18T01:26:09Z","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":"ad00be940917d7c82e97fff5a93486af04fc552dd2006c8cd2fcfd5888ac6f03","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-10-27T22:04:36Z","title_canon_sha256":"bf47e729dcf67d350407f112429c4468b329113a65076413a94732bc71eafe94"},"schema_version":"1.0","source":{"id":"1510.08109","kind":"arxiv","version":2}},"canonical_sha256":"f3eb3aa9a9c92b22f19a9307605c96b63f0d48dd13ec09baeca1984ec2c06d50","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3eb3aa9a9c92b22f19a9307605c96b63f0d48dd13ec09baeca1984ec2c06d50","first_computed_at":"2026-05-18T01:26:09.267926Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:09.267926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XWWlpU2qApo/DTfg95uOKtyNZAIf31GrGZl6iH75+BaV9wYLOApnYha9t/bvY9us8qKXAei99JcX3TDxxWvaDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:09.268401Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.08109","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a5d34aac504c244d740957f4ab63e6f7df2942b0f410f1125ffdec75371195ca","sha256:e96a394eb819aa29dc6fbe079d01c3aade17cb6a919ec14e98c24688d409a133"],"state_sha256":"e0ae85d4b924a731e80585428ca41c892a7eec4b0deb5f427edba89d096c68c3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rPIXASuJKC2mIxImyKWPBAmy/PZ528IFJY0K0GszA8KkQnH6z3OAXkH/+jPf8cHodF1vUsmHIvbmaIsey+PJCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T07:27:34.789511Z","bundle_sha256":"5d5ce4fc8dc5fd8916defa49ea30f89ee403659b7f870e5dbf8d216302346bb7"}}