{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:PM5IFIRCF6YJWKLHEESJHTYS6C","short_pith_number":"pith:PM5IFIRC","canonical_record":{"source":{"id":"2505.01152","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-05-02T09:53:31Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"d2fb5552b24a380862048566121d38b3fec23bdabee4d4e54a7aaa67adb9b902","abstract_canon_sha256":"df916051cf94d2ec514f9b3765006c275804b7982170058106afdc2d25f1fbc1"},"schema_version":"1.0"},"canonical_sha256":"7b3a82a2222fb09b2967212493cf12f08eb42497a366da9c4e3e5b2f659ffed5","source":{"kind":"arxiv","id":"2505.01152","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.01152","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"arxiv_version","alias_value":"2505.01152v2","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.01152","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"pith_short_12","alias_value":"PM5IFIRCF6YJ","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"pith_short_16","alias_value":"PM5IFIRCF6YJWKLH","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"pith_short_8","alias_value":"PM5IFIRC","created_at":"2026-07-05T10:58:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:PM5IFIRCF6YJWKLHEESJHTYS6C","target":"record","payload":{"canonical_record":{"source":{"id":"2505.01152","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-05-02T09:53:31Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"d2fb5552b24a380862048566121d38b3fec23bdabee4d4e54a7aaa67adb9b902","abstract_canon_sha256":"df916051cf94d2ec514f9b3765006c275804b7982170058106afdc2d25f1fbc1"},"schema_version":"1.0"},"canonical_sha256":"7b3a82a2222fb09b2967212493cf12f08eb42497a366da9c4e3e5b2f659ffed5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:58:27.605719Z","signature_b64":"85uew0dMSki/T52RUS1WbC15gR/qz1ct3uwcNuyu2xjOLWLFgixhVzIG0o0mMzi/p9+mvuunUZbAmmmxKgtwAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b3a82a2222fb09b2967212493cf12f08eb42497a366da9c4e3e5b2f659ffed5","last_reissued_at":"2026-07-05T10:58:27.605250Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:58:27.605250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2505.01152","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":"2026-07-05T10:58:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C7TDPNTULJRTi5/bzuceL5PEbBJMNPCBq5/7VmCz6voYSWxaZTqpdduffegoXSaVXQhfuZ1AL6pGDMhujj1oBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T01:15:47.212203Z"},"content_sha256":"0b2d8422bfba3a4f173a17d35a0cc5fcfcca310630e8a16535003e2d71a54f2f","schema_version":"1.0","event_id":"sha256:0b2d8422bfba3a4f173a17d35a0cc5fcfcca310630e8a16535003e2d71a54f2f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:PM5IFIRCF6YJWKLHEESJHTYS6C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polarization Decomposition and Its Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Jun Wang, Tianfu Qi","submitted_at":"2025-05-02T09:53:31Z","abstract_excerpt":"The polarization decomposition of arbitrary binary-input memoryless channels (BMCs) is studied in this work. By introducing the polarization factor (PF), defined in terms of the conditional entropy of the channel output under various input configurations, we demonstrate that the symmetric capacities of the polarized subchannels can be uniformly expressed as functions of the PF. The explicit formulation of the PF as a function of the block length and subchannel index is derived. Furthermore, an efficient algorithm is proposed for the computation of the PF. Notably, we establish a one-to-one cor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01152","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/2505.01152/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":"2026-07-05T10:58:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wc77856OKiAfaGFtosqjnN32b5nFZ56mNmlNSTOuA+orYeQdNj7/RwpJtOZpT5Y6+psaNPXoXoyC8wc2dHYlAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T01:15:47.212717Z"},"content_sha256":"870bc05e0e13e7c82e804cdee9e1ecebb41a240a05aab4218935154c8f0b88fb","schema_version":"1.0","event_id":"sha256:870bc05e0e13e7c82e804cdee9e1ecebb41a240a05aab4218935154c8f0b88fb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PM5IFIRCF6YJWKLHEESJHTYS6C/bundle.json","state_url":"https://pith.science/pith/PM5IFIRCF6YJWKLHEESJHTYS6C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PM5IFIRCF6YJWKLHEESJHTYS6C/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":"2026-08-05T01:15:47Z","links":{"resolver":"https://pith.science/pith/PM5IFIRCF6YJWKLHEESJHTYS6C","bundle":"https://pith.science/pith/PM5IFIRCF6YJWKLHEESJHTYS6C/bundle.json","state":"https://pith.science/pith/PM5IFIRCF6YJWKLHEESJHTYS6C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PM5IFIRCF6YJWKLHEESJHTYS6C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:PM5IFIRCF6YJWKLHEESJHTYS6C","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":"df916051cf94d2ec514f9b3765006c275804b7982170058106afdc2d25f1fbc1","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-05-02T09:53:31Z","title_canon_sha256":"d2fb5552b24a380862048566121d38b3fec23bdabee4d4e54a7aaa67adb9b902"},"schema_version":"1.0","source":{"id":"2505.01152","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.01152","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"arxiv_version","alias_value":"2505.01152v2","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.01152","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"pith_short_12","alias_value":"PM5IFIRCF6YJ","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"pith_short_16","alias_value":"PM5IFIRCF6YJWKLH","created_at":"2026-07-05T10:58:27Z"},{"alias_kind":"pith_short_8","alias_value":"PM5IFIRC","created_at":"2026-07-05T10:58:27Z"}],"graph_snapshots":[{"event_id":"sha256:870bc05e0e13e7c82e804cdee9e1ecebb41a240a05aab4218935154c8f0b88fb","target":"graph","created_at":"2026-07-05T10:58:27Z","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/2505.01152/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The polarization decomposition of arbitrary binary-input memoryless channels (BMCs) is studied in this work. By introducing the polarization factor (PF), defined in terms of the conditional entropy of the channel output under various input configurations, we demonstrate that the symmetric capacities of the polarized subchannels can be uniformly expressed as functions of the PF. The explicit formulation of the PF as a function of the block length and subchannel index is derived. Furthermore, an efficient algorithm is proposed for the computation of the PF. Notably, we establish a one-to-one cor","authors_text":"Jun Wang, Tianfu Qi","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-05-02T09:53:31Z","title":"Polarization Decomposition and Its Applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.01152","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":"sha256:0b2d8422bfba3a4f173a17d35a0cc5fcfcca310630e8a16535003e2d71a54f2f","target":"record","created_at":"2026-07-05T10:58:27Z","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":"df916051cf94d2ec514f9b3765006c275804b7982170058106afdc2d25f1fbc1","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-05-02T09:53:31Z","title_canon_sha256":"d2fb5552b24a380862048566121d38b3fec23bdabee4d4e54a7aaa67adb9b902"},"schema_version":"1.0","source":{"id":"2505.01152","kind":"arxiv","version":2}},"canonical_sha256":"7b3a82a2222fb09b2967212493cf12f08eb42497a366da9c4e3e5b2f659ffed5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b3a82a2222fb09b2967212493cf12f08eb42497a366da9c4e3e5b2f659ffed5","first_computed_at":"2026-07-05T10:58:27.605250Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:58:27.605250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"85uew0dMSki/T52RUS1WbC15gR/qz1ct3uwcNuyu2xjOLWLFgixhVzIG0o0mMzi/p9+mvuunUZbAmmmxKgtwAg==","signature_status":"signed_v1","signed_at":"2026-07-05T10:58:27.605719Z","signed_message":"canonical_sha256_bytes"},"source_id":"2505.01152","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b2d8422bfba3a4f173a17d35a0cc5fcfcca310630e8a16535003e2d71a54f2f","sha256:870bc05e0e13e7c82e804cdee9e1ecebb41a240a05aab4218935154c8f0b88fb"],"state_sha256":"79d8fc9884a7faabede75016b26eff06e408eccd3085b2e48ab5131b42d2a3a0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CCuJG0vW11SS298wUn3XUL1cCt9ApN/Z2ZRbmov6qpZbGakeOP3RZAPsPbvNabeQSA4hm0p9zoQsYtbsgBEfCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T01:15:47.216137Z","bundle_sha256":"e262cbf92469aebc954965d1e7d04d227aa58ea16ae68e7c7cec0e4360bf68dc"}}