{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:ASIN4PMIKWVYPLU75OJJDQMIHR","short_pith_number":"pith:ASIN4PMI","canonical_record":{"source":{"id":"1206.4542","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-19T17:32:31Z","cross_cats_sorted":[],"title_canon_sha256":"c740da3d0c1d65aabdf61a220e44f206a76764df063581eae1289ebfb196e583","abstract_canon_sha256":"53322fefb0dbc2aefae78cfa53f12f0cfcad89b0d63c20749bb3d0c314787553"},"schema_version":"1.0"},"canonical_sha256":"0490de3d8855ab87ae9feb9291c1883c7fbccbcdfea58298529f5cc5c83b62b2","source":{"kind":"arxiv","id":"1206.4542","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4542","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4542v2","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4542","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"pith_short_12","alias_value":"ASIN4PMIKWVY","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"ASIN4PMIKWVYPLU7","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"ASIN4PMI","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:ASIN4PMIKWVYPLU75OJJDQMIHR","target":"record","payload":{"canonical_record":{"source":{"id":"1206.4542","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-19T17:32:31Z","cross_cats_sorted":[],"title_canon_sha256":"c740da3d0c1d65aabdf61a220e44f206a76764df063581eae1289ebfb196e583","abstract_canon_sha256":"53322fefb0dbc2aefae78cfa53f12f0cfcad89b0d63c20749bb3d0c314787553"},"schema_version":"1.0"},"canonical_sha256":"0490de3d8855ab87ae9feb9291c1883c7fbccbcdfea58298529f5cc5c83b62b2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:38.468311Z","signature_b64":"MEeQpk6DT0HA2LmCTvjSzsSUX7gOSlNRma4v9ncufNUdbQit1A+tepL93BgRIPwsd8uJfhNunlxDqXp1zHtvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0490de3d8855ab87ae9feb9291c1883c7fbccbcdfea58298529f5cc5c83b62b2","last_reissued_at":"2026-05-18T03:35:38.467402Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:38.467402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.4542","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":"2026-05-18T03:35:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rcfdi03zIZSbu/jRilzT37XQX5oo5476O7pcUKhW4IQRhZTl/g6/I0rZAg6koL0BXYJUbDcmN/g3sJQwiKK3Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:50:21.686735Z"},"content_sha256":"8692dc49688ac1f0be1a0669f8b9033c6440f4cad2ed0aeef36bdf65ed590f4b","schema_version":"1.0","event_id":"sha256:8692dc49688ac1f0be1a0669f8b9033c6440f4cad2ed0aeef36bdf65ed590f4b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:ASIN4PMIKWVYPLU75OJJDQMIHR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Infinite Dimensional Bicomplex Spectral Decomposition Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dominic Rochon, Kuldeep Singh Charak, Ravinder Kumar","submitted_at":"2012-06-19T17:32:31Z","abstract_excerpt":"This paper presents a bicomplex version of the Spectral Decomposition Theorem on infinite dimensional bicomplex Hilbert spaces. In the process, the ideas of bounded linear operators, orthogonal complements and compact operators on bicomplex Hilbert spaces are introduced and treated in relation with the classical Hilbert space $M'$ imbedded in any bicomplex Hilbert space $M$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4542","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":"2026-05-18T03:35:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wmPOpTmTEWEutaQL7zpisNHTZRi9FqIT9RJqRuCUaVtGMoem8v5t1s/PWW46O0T6Vt4nYcWroW2x4F+c/r00Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T04:50:21.687057Z"},"content_sha256":"6f55ba0aae901a22192c77933a4f988210f718e00b30b4d3e6eed14ee38b7d2c","schema_version":"1.0","event_id":"sha256:6f55ba0aae901a22192c77933a4f988210f718e00b30b4d3e6eed14ee38b7d2c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ASIN4PMIKWVYPLU75OJJDQMIHR/bundle.json","state_url":"https://pith.science/pith/ASIN4PMIKWVYPLU75OJJDQMIHR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ASIN4PMIKWVYPLU75OJJDQMIHR/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":"2026-06-27T04:50:21Z","links":{"resolver":"https://pith.science/pith/ASIN4PMIKWVYPLU75OJJDQMIHR","bundle":"https://pith.science/pith/ASIN4PMIKWVYPLU75OJJDQMIHR/bundle.json","state":"https://pith.science/pith/ASIN4PMIKWVYPLU75OJJDQMIHR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ASIN4PMIKWVYPLU75OJJDQMIHR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ASIN4PMIKWVYPLU75OJJDQMIHR","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":"53322fefb0dbc2aefae78cfa53f12f0cfcad89b0d63c20749bb3d0c314787553","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-19T17:32:31Z","title_canon_sha256":"c740da3d0c1d65aabdf61a220e44f206a76764df063581eae1289ebfb196e583"},"schema_version":"1.0","source":{"id":"1206.4542","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4542","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4542v2","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4542","created_at":"2026-05-18T03:35:38Z"},{"alias_kind":"pith_short_12","alias_value":"ASIN4PMIKWVY","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"ASIN4PMIKWVYPLU7","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"ASIN4PMI","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:6f55ba0aae901a22192c77933a4f988210f718e00b30b4d3e6eed14ee38b7d2c","target":"graph","created_at":"2026-05-18T03:35:38Z","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":"This paper presents a bicomplex version of the Spectral Decomposition Theorem on infinite dimensional bicomplex Hilbert spaces. In the process, the ideas of bounded linear operators, orthogonal complements and compact operators on bicomplex Hilbert spaces are introduced and treated in relation with the classical Hilbert space $M'$ imbedded in any bicomplex Hilbert space $M$.","authors_text":"Dominic Rochon, Kuldeep Singh Charak, Ravinder Kumar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-19T17:32:31Z","title":"Infinite Dimensional Bicomplex Spectral Decomposition Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4542","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":"sha256:8692dc49688ac1f0be1a0669f8b9033c6440f4cad2ed0aeef36bdf65ed590f4b","target":"record","created_at":"2026-05-18T03:35:38Z","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":"53322fefb0dbc2aefae78cfa53f12f0cfcad89b0d63c20749bb3d0c314787553","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-19T17:32:31Z","title_canon_sha256":"c740da3d0c1d65aabdf61a220e44f206a76764df063581eae1289ebfb196e583"},"schema_version":"1.0","source":{"id":"1206.4542","kind":"arxiv","version":2}},"canonical_sha256":"0490de3d8855ab87ae9feb9291c1883c7fbccbcdfea58298529f5cc5c83b62b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0490de3d8855ab87ae9feb9291c1883c7fbccbcdfea58298529f5cc5c83b62b2","first_computed_at":"2026-05-18T03:35:38.467402Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:35:38.467402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MEeQpk6DT0HA2LmCTvjSzsSUX7gOSlNRma4v9ncufNUdbQit1A+tepL93BgRIPwsd8uJfhNunlxDqXp1zHtvAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:35:38.468311Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.4542","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8692dc49688ac1f0be1a0669f8b9033c6440f4cad2ed0aeef36bdf65ed590f4b","sha256:6f55ba0aae901a22192c77933a4f988210f718e00b30b4d3e6eed14ee38b7d2c"],"state_sha256":"eec72567a7e1649bd34f17fede18c33bbe581a9ff0d31b381e819c4a65052cc1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RqY4K5pGSWCzZFPqP5PZmEaevXNimfCy3UkU/n70Qc3yrkZ6IS8Azq08KoHHdJCiLab2hzWMIvtPgtay5Mj2CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T04:50:21.688769Z","bundle_sha256":"b483a3d8668ffb3e76decc6228f38624aec548799451bcec443fadff8f34f4b3"}}