{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:432RWOZ4F37GDR4SEUAZG7F4YH","short_pith_number":"pith:432RWOZ4","canonical_record":{"source":{"id":"0809.1573","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2008-09-09T15:07:55Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d111a8b111781c6710e13f997c47ecadc51fe7e775c26bdc2ea53eb07418fef4","abstract_canon_sha256":"3c7062a3637b4aef2c8c571a66f3ae1a29917c22a57ca962f828511fe5193689"},"schema_version":"1.0"},"canonical_sha256":"e6f51b3b3c2efe61c7922501937cbcc1c57b444a76b4896e53443a0c561ec306","source":{"kind":"arxiv","id":"0809.1573","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.1573","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"arxiv_version","alias_value":"0809.1573v2","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.1573","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"pith_short_12","alias_value":"432RWOZ4F37G","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"432RWOZ4F37GDR4S","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"432RWOZ4","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:432RWOZ4F37GDR4SEUAZG7F4YH","target":"record","payload":{"canonical_record":{"source":{"id":"0809.1573","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2008-09-09T15:07:55Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d111a8b111781c6710e13f997c47ecadc51fe7e775c26bdc2ea53eb07418fef4","abstract_canon_sha256":"3c7062a3637b4aef2c8c571a66f3ae1a29917c22a57ca962f828511fe5193689"},"schema_version":"1.0"},"canonical_sha256":"e6f51b3b3c2efe61c7922501937cbcc1c57b444a76b4896e53443a0c561ec306","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:17.083192Z","signature_b64":"dFz0IdcFNSHzxJxWsXzedy37uwhmC2isqIH4cgoxFAN98xRQ2KXcVCThzijMuGtrssocbdhh1NC7IbADJhE3DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6f51b3b3c2efe61c7922501937cbcc1c57b444a76b4896e53443a0c561ec306","last_reissued_at":"2026-05-18T04:39:17.082699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:17.082699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0809.1573","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":"2026-05-18T04:39:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cim5KMFYoCAMU48uLSnvbiHxPq9yE3aZinUx//YQRxaMuzusjosfhdLMdoAO85PUhIeVivaE3N08YGlc+2t3Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T01:13:50.734962Z"},"content_sha256":"6541c0e12ab3df0d38fd777f17850c49442b2fe23b6ccf5400eb5d355aa9ec62","schema_version":"1.0","event_id":"sha256:6541c0e12ab3df0d38fd777f17850c49442b2fe23b6ccf5400eb5d355aa9ec62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:432RWOZ4F37GDR4SEUAZG7F4YH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stabilization in $H^\\infty_{\\mathbb{R}}(\\mathbb{D})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CV","authors_text":"Brett D. Wick","submitted_at":"2008-09-09T15:07:55Z","abstract_excerpt":"In this paper we prove the following theorem: Suppose that $f_1,f_2\\in H^\\infty_\\R(\\D)$, with $\\norm{f_1}_\\infty,\\norm{f_2}_{\\infty}\\leq 1$, with $$ \\inf_{z\\in\\D}(\\abs{f_1(z)}+\\abs{f_2(z)})=\\delta>0. $$ Assume for some $\\epsilon>0$ and small, $f_1$ is positive on the set of $x\\in(-1,1)$ where $\\abs{f_2(x)}<\\epsilon$ for some $\\epsilon>0$ sufficiently small. Then there exists $g_1, g_1^{-1}, g_2\\in H^\\infty_\\R(\\D)$ with $$ \\norm{g_1}_\\infty,\\norm{g_2}_\\infty,\\norm{g_1^{-1}}_\\infty\\leq C(\\delta,\\epsilon) $$ and $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\\quad\\forall z\\in\\D. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.1573","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":"2026-05-18T04:39:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"raYGrQzVx8Q5yQbP6jN0rYVGSRyNlTq+gWD3jBsUCJJD2EOmkwMj/v343gfc1/i6A4xNEvKGEgbo+EjoFB+aAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T01:13:50.735325Z"},"content_sha256":"ad74d0cddd827bcf091f178d43291a4e0607b4d60503c29c3bc1737b0cb92cba","schema_version":"1.0","event_id":"sha256:ad74d0cddd827bcf091f178d43291a4e0607b4d60503c29c3bc1737b0cb92cba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/432RWOZ4F37GDR4SEUAZG7F4YH/bundle.json","state_url":"https://pith.science/pith/432RWOZ4F37GDR4SEUAZG7F4YH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/432RWOZ4F37GDR4SEUAZG7F4YH/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":"2026-06-01T01:13:50Z","links":{"resolver":"https://pith.science/pith/432RWOZ4F37GDR4SEUAZG7F4YH","bundle":"https://pith.science/pith/432RWOZ4F37GDR4SEUAZG7F4YH/bundle.json","state":"https://pith.science/pith/432RWOZ4F37GDR4SEUAZG7F4YH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/432RWOZ4F37GDR4SEUAZG7F4YH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:432RWOZ4F37GDR4SEUAZG7F4YH","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":"3c7062a3637b4aef2c8c571a66f3ae1a29917c22a57ca962f828511fe5193689","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2008-09-09T15:07:55Z","title_canon_sha256":"d111a8b111781c6710e13f997c47ecadc51fe7e775c26bdc2ea53eb07418fef4"},"schema_version":"1.0","source":{"id":"0809.1573","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.1573","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"arxiv_version","alias_value":"0809.1573v2","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.1573","created_at":"2026-05-18T04:39:17Z"},{"alias_kind":"pith_short_12","alias_value":"432RWOZ4F37G","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"432RWOZ4F37GDR4S","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"432RWOZ4","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:ad74d0cddd827bcf091f178d43291a4e0607b4d60503c29c3bc1737b0cb92cba","target":"graph","created_at":"2026-05-18T04:39:17Z","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 this paper we prove the following theorem: Suppose that $f_1,f_2\\in H^\\infty_\\R(\\D)$, with $\\norm{f_1}_\\infty,\\norm{f_2}_{\\infty}\\leq 1$, with $$ \\inf_{z\\in\\D}(\\abs{f_1(z)}+\\abs{f_2(z)})=\\delta>0. $$ Assume for some $\\epsilon>0$ and small, $f_1$ is positive on the set of $x\\in(-1,1)$ where $\\abs{f_2(x)}<\\epsilon$ for some $\\epsilon>0$ sufficiently small. Then there exists $g_1, g_1^{-1}, g_2\\in H^\\infty_\\R(\\D)$ with $$ \\norm{g_1}_\\infty,\\norm{g_2}_\\infty,\\norm{g_1^{-1}}_\\infty\\leq C(\\delta,\\epsilon) $$ and $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\\quad\\forall z\\in\\D. $$","authors_text":"Brett D. Wick","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2008-09-09T15:07:55Z","title":"Stabilization in $H^\\infty_{\\mathbb{R}}(\\mathbb{D})$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.1573","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":"sha256:6541c0e12ab3df0d38fd777f17850c49442b2fe23b6ccf5400eb5d355aa9ec62","target":"record","created_at":"2026-05-18T04:39:17Z","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":"3c7062a3637b4aef2c8c571a66f3ae1a29917c22a57ca962f828511fe5193689","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2008-09-09T15:07:55Z","title_canon_sha256":"d111a8b111781c6710e13f997c47ecadc51fe7e775c26bdc2ea53eb07418fef4"},"schema_version":"1.0","source":{"id":"0809.1573","kind":"arxiv","version":2}},"canonical_sha256":"e6f51b3b3c2efe61c7922501937cbcc1c57b444a76b4896e53443a0c561ec306","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6f51b3b3c2efe61c7922501937cbcc1c57b444a76b4896e53443a0c561ec306","first_computed_at":"2026-05-18T04:39:17.082699Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:17.082699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dFz0IdcFNSHzxJxWsXzedy37uwhmC2isqIH4cgoxFAN98xRQ2KXcVCThzijMuGtrssocbdhh1NC7IbADJhE3DA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:17.083192Z","signed_message":"canonical_sha256_bytes"},"source_id":"0809.1573","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6541c0e12ab3df0d38fd777f17850c49442b2fe23b6ccf5400eb5d355aa9ec62","sha256:ad74d0cddd827bcf091f178d43291a4e0607b4d60503c29c3bc1737b0cb92cba"],"state_sha256":"cc487b3dfb5e93b3b9b90525357638b5ab0cf49e45edebf86f5510160014d8aa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BGZN7LA3iHe4KJKoy9Mra6DyAfnVIau039tkyhFi5pEo4D7ujpzay1EIf7TFoIQmL5+pqeJj+Y74O+eNr+EbBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T01:13:50.737686Z","bundle_sha256":"4d65c90ead24605332d58ce140023c80d27dd2ff4c8f6d2e7441a69f145925a8"}}