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We decompose 1776 heads across five pretrained transformers and find routing operating at low rank, well below the routing capacity allocated by the weight kernel. We introduce $S$-$D$ attention as a diagnostic parameterization that disentangles routing from filtering by construction with guaranteed stability ($\\mathrm{Re}(\\lambda) \\le 0$) and trains stably without layer normalization. When disentangled and unnormalized, routing self-organizes into a spectral cascade, effective rank $2$ at the first layer, expanding with depth across six scales from 7M to 355M parameters. The cascade predicts where attention can be simplified: linearizing the first seven layers of 125M $S$-$D$ attention costs ${<}5\\%$ pe","title":"The Routing and Filtering Structure of Attention","url":"https://arxiv.org/abs/2605.18826","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.18826v1 Announce Type: cross \nAbstract: The attention interaction matrix $QK^{\\top}$ contains two entangled computations: a skew-symmetric component that redistributes information between positions (routing) and a symmetric component that scales mutual relevance (filtering). We decompose 1776 heads across five pretrained transformers and find routing operating at low rank, well below the routing capacity allocated by the weight kernel. We introduce $S$-$D$ attention as a diagnostic parameterization that disentangles routing from filtering by construction with guaranteed stability ($\\mathrm{Re}(\\lambda) \\le 0$) and trains stably without layer normalization. When disentangled and unnormalized, routing self-organizes into a spectral cascade, effective rank $2$ at the first layer, expanding with depth across six scales from 7M to 355M parameters. The cascade predicts where attention can be simplified: linearizing the first seven layers of 125M $S$-$D$ attention costs ${<}5\\%$ pe","title":"The Routing and Filtering Structure of Attention","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-20T04:43:44Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.18826"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:5c523ef1166656847e3721cc1e19bd373db73e53f6c180529c24806a7dbdc3607d01da353a81d057cf2b82f077d393075d79b20f9024dda8dd2c0ab7837eea05","signer":"crovia.substrate","subject":{"observed_at":"2026-05-20T04:43:44Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.18826"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"5e349ee525a898957ce2ea9200ac3bc41ebf550b95354895f3090c6d153b9f02","leaf_index":145005,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"ea2042859b889624d433acc0ec9c8fa669d30297d18ea55c6c22d158ba821e2d","side":"left"},{"sibling":"92c54a2e79fc54fa03a6669a827543a798a71a24575f94e9e99185e5fc467768","side":"right"},{"sibling":"4893adf01de5e3afe733d49d52b5bcdb973ab0eaf5b376963273912b9b547610","side":"left"},{"sibling":"0bcbcbd7c737ea88fcc2bef8cda93284e723aeb148ed75ec519cf076c7e1ad0c","side":"left"},{"sibling":"e4d6610283bc29c91371d0a15e64ff55b0a39d2964943ab4137f82d52813670d","side":"right"},{"sibling":"5efc60a8437bb8dfc7d20976dabe21eacc357a19aa672a53914e2792ef2fa629","side":"left"},{"sibling":"41f0af342418ef1081b5d968b56fe8290cf77b51073587d5abca55a1dcf5d23b","side":"left"},{"sibling":"5c26ec91378ced2564bd7218068bacda3ade3a0517e2dc081e9d169a340044ba","side":"right"},{"sibling":"9db04d2181fb869f16a5baf7dcfe999cc93a7e9beef8493d0cc71f57470fe5e1","side":"right"},{"sibling":"1526885f19d1fadf6955cf519dbc4e62d593a4bba99d741e8c301740a7068233","side":"left"},{"sibling":"e3a7d5c07f161682d61bd453ffc02ecdf87cfeda70f986d6650017f9d2d6b265","side":"left"},{"sibling":"edbc49f08e5b92291934c05c9e6efd270a6b0698d8d2fa474006366027dfe098","side":"right"},{"sibling":"3e4df6e7457cecbf36f350375e72dcab336a3984422e4c406ef809e4e2944e96","side":"left"},{"sibling":"8f4c0fbe56b6c010fbb8c782ebcd478079bb3f991d8704e2534209a075d9163c","side":"left"},{"sibling":"be08fedc4e72a6fb56606f66812fae7317e09690b9acb18385f4ab117a981238","side":"right"},{"sibling":"0534329a7475dc9df51998c83c16892126679dade0fa34182f21e869599386c7","side":"right"},{"sibling":"2d24720928ead0e7670650eb55f558c4f20e4c18df376f47ba72cfa8cf0ed344","side":"right"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":147301,"merkle_root":"08903d7159c3b38eeeeafc09eab15139ea417f1d94f02f1fbc87296b37db840a","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260521T183701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-21T18:37:33Z","sig_algorithm":"ed25519","signature":"905f2924632dfa2970c8690285f5b5d4a1d891d0e0ef1cbc404ebec2fd937215ac768e16f0a9f28b18977a55ae0bfd226db6833ae7ef588729054117d2da7303","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_9f1c4eea8e5bb8ba101ff78122e50f50560909ee8ba8197eb03a77697423ea28"}}