{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:IJ34C657MPUBJUCNSJ2AWZMCGL","short_pith_number":"pith:IJ34C657","canonical_record":{"source":{"id":"1010.0565","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-04T12:34:53Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"60485140cb45acc554acb4b3329e3fd94bf7fb498a31020f80fdb9efdf10772f","abstract_canon_sha256":"a1c8c9711a2c32219828c5adaf1ca0d0af5a763cef208e14ccd4007beaafa083"},"schema_version":"1.0"},"canonical_sha256":"4277c17bbf63e814d04d92740b658232e083fa0c42979e574619734bed4714e2","source":{"kind":"arxiv","id":"1010.0565","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.0565","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"arxiv_version","alias_value":"1010.0565v1","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.0565","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"pith_short_12","alias_value":"IJ34C657MPUB","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"IJ34C657MPUBJUCN","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"IJ34C657","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:IJ34C657MPUBJUCNSJ2AWZMCGL","target":"record","payload":{"canonical_record":{"source":{"id":"1010.0565","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-04T12:34:53Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"60485140cb45acc554acb4b3329e3fd94bf7fb498a31020f80fdb9efdf10772f","abstract_canon_sha256":"a1c8c9711a2c32219828c5adaf1ca0d0af5a763cef208e14ccd4007beaafa083"},"schema_version":"1.0"},"canonical_sha256":"4277c17bbf63e814d04d92740b658232e083fa0c42979e574619734bed4714e2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:51.806071Z","signature_b64":"3LxGgmcHbZzcwrltAd34waX9X2vRWzLFN4uIPz1US858ETO53b0i8H+L5XLwzx5kvFa7j3HecCd9lAr1pZAqDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4277c17bbf63e814d04d92740b658232e083fa0c42979e574619734bed4714e2","last_reissued_at":"2026-05-18T04:39:51.805413Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:51.805413Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.0565","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":"2026-05-18T04:39:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UUj73caXY3AWaLhJedZ01b6ULgF7U96ApcOa1jEx/a4ydyx7tAvHuV7jBtPYE45ciP67P9vvLdu1ywMZWSURDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T03:26:22.796177Z"},"content_sha256":"52222092c7198eb6890ec6864bf91b3a9c37a175dbf2dd15e9623ce494fb8dfb","schema_version":"1.0","event_id":"sha256:52222092c7198eb6890ec6864bf91b3a9c37a175dbf2dd15e9623ce494fb8dfb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:IJ34C657MPUBJUCNSJ2AWZMCGL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Ulam stability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.FA","authors_text":"Andreas Thom, Marc Burger, Narutaka Ozawa","submitted_at":"2010-10-04T12:34:53Z","abstract_excerpt":"We study $\\epsilon$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $\\epsilon$-represendations are uniformly close to unitary representations. One of our main results is that certain lattices in connected semi-simple Lie groups of higher rank are Ulam stable. For infinite-dimensional $\\epsilon$-representations, the similarly defined notion of strong Ulam stability is defined and it is shown that groups with free subgroups are not strongly Ulam stable. We also study deformat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.0565","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":"2026-05-18T04:39:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mjV66YPQGz8XXm7gCT/JySM9ISHJONhzNhF0OEqf1pcz9Vro2h9oUI8fLqnafKVBYZBPaZCQLUhfvuc+ib6kCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T03:26:22.796556Z"},"content_sha256":"a7d9a102acf7ca5026595d509843605a10bda2e8e9fec762790776f3f1719882","schema_version":"1.0","event_id":"sha256:a7d9a102acf7ca5026595d509843605a10bda2e8e9fec762790776f3f1719882"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IJ34C657MPUBJUCNSJ2AWZMCGL/bundle.json","state_url":"https://pith.science/pith/IJ34C657MPUBJUCNSJ2AWZMCGL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IJ34C657MPUBJUCNSJ2AWZMCGL/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":"2026-05-28T03:26:22Z","links":{"resolver":"https://pith.science/pith/IJ34C657MPUBJUCNSJ2AWZMCGL","bundle":"https://pith.science/pith/IJ34C657MPUBJUCNSJ2AWZMCGL/bundle.json","state":"https://pith.science/pith/IJ34C657MPUBJUCNSJ2AWZMCGL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IJ34C657MPUBJUCNSJ2AWZMCGL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:IJ34C657MPUBJUCNSJ2AWZMCGL","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":"a1c8c9711a2c32219828c5adaf1ca0d0af5a763cef208e14ccd4007beaafa083","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-04T12:34:53Z","title_canon_sha256":"60485140cb45acc554acb4b3329e3fd94bf7fb498a31020f80fdb9efdf10772f"},"schema_version":"1.0","source":{"id":"1010.0565","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.0565","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"arxiv_version","alias_value":"1010.0565v1","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.0565","created_at":"2026-05-18T04:39:51Z"},{"alias_kind":"pith_short_12","alias_value":"IJ34C657MPUB","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"IJ34C657MPUBJUCN","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"IJ34C657","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:a7d9a102acf7ca5026595d509843605a10bda2e8e9fec762790776f3f1719882","target":"graph","created_at":"2026-05-18T04:39:51Z","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 study $\\epsilon$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $\\epsilon$-represendations are uniformly close to unitary representations. One of our main results is that certain lattices in connected semi-simple Lie groups of higher rank are Ulam stable. For infinite-dimensional $\\epsilon$-representations, the similarly defined notion of strong Ulam stability is defined and it is shown that groups with free subgroups are not strongly Ulam stable. We also study deformat","authors_text":"Andreas Thom, Marc Burger, Narutaka Ozawa","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-04T12:34:53Z","title":"On Ulam stability"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.0565","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":"sha256:52222092c7198eb6890ec6864bf91b3a9c37a175dbf2dd15e9623ce494fb8dfb","target":"record","created_at":"2026-05-18T04:39:51Z","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":"a1c8c9711a2c32219828c5adaf1ca0d0af5a763cef208e14ccd4007beaafa083","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-04T12:34:53Z","title_canon_sha256":"60485140cb45acc554acb4b3329e3fd94bf7fb498a31020f80fdb9efdf10772f"},"schema_version":"1.0","source":{"id":"1010.0565","kind":"arxiv","version":1}},"canonical_sha256":"4277c17bbf63e814d04d92740b658232e083fa0c42979e574619734bed4714e2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4277c17bbf63e814d04d92740b658232e083fa0c42979e574619734bed4714e2","first_computed_at":"2026-05-18T04:39:51.805413Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:51.805413Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3LxGgmcHbZzcwrltAd34waX9X2vRWzLFN4uIPz1US858ETO53b0i8H+L5XLwzx5kvFa7j3HecCd9lAr1pZAqDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:51.806071Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.0565","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52222092c7198eb6890ec6864bf91b3a9c37a175dbf2dd15e9623ce494fb8dfb","sha256:a7d9a102acf7ca5026595d509843605a10bda2e8e9fec762790776f3f1719882"],"state_sha256":"41e49a8da269524013afad84e4f29d1d2cba488cca58229bd9ee27948004e642"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4rXOyl0NWXbuXsjW5DZdHnCsbbW+N3SNi9YaahVR0+jjs6ps4sT2XLPNVy0F3wteANaMxWMIWr1l6gIfNgHBCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T03:26:22.798559Z","bundle_sha256":"1c31f8adaff470e260f207c887d7cecb8caf56e0ee14a9272cc3bdb3e460e242"}}