{"paper":{"title":"Complex Analysis of Channel Polarization on Discrete BMS Channels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Complex analysis yields exact Bhattacharyya parameters for polarized bit-channels on any discrete BMS channel.","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Dongxiao Xu, Holger Boche, Moritz Wiese","submitted_at":"2026-05-05T14:31:11Z","abstract_excerpt":"We develop component evolution (CE), a framework based on complex function theory for finite-blocklength channel polarization on discrete binary-input memoryless output-symmetric (BMS) channels. In this view, the Bhattacharyya parameter is treated as a real-valued instance of a broader class of complex-valued channel functionals. CE systematically derives analytic expressions for the Bhattacharyya parameters of the bit-channels of a given discrete BMS channel at arbitrary polarization levels. CE also enables structural analysis, providing new evidence of extremality of the binary erasure chann"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"CE systematically derives analytic expressions for the Bhattacharyya parameters of the bit-channels of a given discrete BMS channel at arbitrary polarization levels.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Bhattacharyya parameter admits a natural and useful extension to a broader class of complex-valued channel functionals that preserves the key polarization properties and allows closed-form recursion.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new complex-analytic framework derives analytic expressions for Bhattacharyya parameters of polarized bit-channels on discrete BMS channels and provides evidence for extremality of BEC and BSC.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Complex analysis yields exact Bhattacharyya parameters for polarized bit-channels on any discrete BMS channel.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bef50ce2ea9267ed99bd7ec982363e8995699d929c04b94ae9c77693e35fef2f"},"source":{"id":"2605.03805","kind":"arxiv","version":2},"verdict":{"id":"6cf945ed-9501-4f03-8090-557bed7b95ff","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T13:32:52.553220Z","strongest_claim":"CE systematically derives analytic expressions for the Bhattacharyya parameters of the bit-channels of a given discrete BMS channel at arbitrary polarization levels.","one_line_summary":"A new complex-analytic framework derives analytic expressions for Bhattacharyya parameters of polarized bit-channels on discrete BMS channels and provides evidence for extremality of BEC and BSC.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Bhattacharyya parameter admits a natural and useful extension to a broader class of complex-valued channel functionals that preserves the key polarization properties and allows closed-form recursion.","pith_extraction_headline":"Complex analysis yields exact Bhattacharyya parameters for polarized bit-channels on any discrete BMS channel."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.03805/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T15:01:55.869809Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"963685f3e784c2d5899239d05bb66fa83a3049462ab2b736dc68f8a8ac067a48"},"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"}