{"paper":{"title":"Jordan-RoPE: Non-Semisimple Relative Positional Encoding via Complex Jordan Blocks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Non-semisimple Jordan blocks realize a distance-modulated phase basis for relative positional encoding","cross_cats":["cs.CL"],"primary_cat":"cs.LG","authors_text":"Yaobo Zhang","submitted_at":"2026-05-05T18:59:31Z","abstract_excerpt":"Relative positional encodings determine which functions of query-key lag can enter the primitive attention logit. RoPE supplies a rotary phase, while ALiBi supplies an additive distance bias. Motivated by group-theoretic views of linear translation-invariant positional encodings, we study a non-semisimple case in which a complex rotary eigenvalue and a nilpotent response live in the same defective Jordan block. The resulting relative operator generates oscillatory-polynomial features such as $e^{-\\gamma d}\\cos(\\omega d)$, $e^{-\\gamma d}\\sin(\\omega d)$, $d e^{-\\gamma d}\\cos(\\omega d)$, and $d e"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the construction realizes a distance-modulated phase basis d e^{iωd}, rather than merely adding a separate distance channel to RoPE.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the non-semisimple Jordan structure supplies an inductive bias that cannot be matched by simply combining existing RoPE and ALiBi channels or by other stabilized variants.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Jordan-RoPE realizes a non-semisimple relative positional operator that produces coupled oscillatory-polynomial features such as d e^{i omega d} for causal query-key lags.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Non-semisimple Jordan blocks realize a distance-modulated phase basis for relative positional encoding","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fc711af3fe77e3f5d8d18db2ddede5114ade7bebaddba6761f191cf3001abb36"},"source":{"id":"2605.04217","kind":"arxiv","version":2},"verdict":{"id":"45d65000-3658-495c-9334-7e7e5f7d554b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T17:06:24.228375Z","strongest_claim":"the construction realizes a distance-modulated phase basis d e^{iωd}, rather than merely adding a separate distance channel to RoPE.","one_line_summary":"Jordan-RoPE realizes a non-semisimple relative positional operator that produces coupled oscillatory-polynomial features such as d e^{i omega d} for causal query-key lags.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the non-semisimple Jordan structure supplies an inductive bias that cannot be matched by simply combining existing RoPE and ALiBi channels or by other stabilized variants.","pith_extraction_headline":"Non-semisimple Jordan blocks realize a distance-modulated phase basis for relative positional encoding"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04217/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T12:36:24.202863Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:21.071369Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:42:10.479550Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"edf6528cc2f93bc21de25d58173ff526ff19dc9ed8456313cceb427b27c84dfa"},"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"}