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Motivated by empirical work on high-learning-rate transformer instabilities and by the cubic-map phase diagram for quadratic regression, we study an exactly reducible one-prompt linear-transformer training problem. After normalization, the dynamics reduce to a two-factor product map with an effective step-size parameter \\(\\mu\\). On the balanced slice, this map recovers the known scalar cubic transition from monotone convergence to catapult convergence, periodic and chaotic bounded nonconvergence, and divergence. We then analyze the full two-dimensional system and show that, for \\(0<\\mu<2\\), it has an explicit invariant Chebyshev ellipse separating forward-invariant regions; this ellipse carries off-balanced chaotic dynamics but","title":"Large-Step Training Dynamics of a Two-Factor Linear Transformer Model","url":"https://arxiv.org/abs/2605.21292","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.21292v1 Announce Type: cross \nAbstract: Gradient-flow analyses show that simplified linear transformers can learn the in-context linear-regression algorithm, but they do not explain the finite-step behavior of gradient descent at large learning rates. Motivated by empirical work on high-learning-rate transformer instabilities and by the cubic-map phase diagram for quadratic regression, we study an exactly reducible one-prompt linear-transformer training problem. After normalization, the dynamics reduce to a two-factor product map with an effective step-size parameter \\(\\mu\\). On the balanced slice, this map recovers the known scalar cubic transition from monotone convergence to catapult convergence, periodic and chaotic bounded nonconvergence, and divergence. We then analyze the full two-dimensional system and show that, for \\(0<\\mu<2\\), it has an explicit invariant Chebyshev ellipse separating forward-invariant regions; this ellipse carries off-balanced chaotic dynamics but","title":"Large-Step Training Dynamics of a Two-Factor Linear Transformer Model","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-22T04:43:12Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.21292"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:80139a4e0afe254e551381148095414711254844d1be48917ec077d82967af60d26225fac766a035d30df9db64d1bfefbb8d7cba39924ee8395783807368a201","signer":"crovia.substrate","subject":{"observed_at":"2026-05-22T04:43:12Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.21292"},"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":"ea40ec5e310e4565c16a075d7482dd8e1d7f7f6df96762002c58bebbe93d7517","leaf_index":148329,"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":"e5c0572dd95292ae6bce6f23b27379e28f6e5b1263cafa1b696f246fca283774","side":"left"},{"sibling":"e7f6d222a84d30d58cf1c42bcc2bbbe8e30c2af76900bae3dd8bebf7354deb24","side":"right"},{"sibling":"532417737b2c9fce5da50352864bf3347911e0ebfe8829baab4b6bf29a8cda4a","side":"right"},{"sibling":"d195b43cfebbc4991a979580fc17c6eba01c871316e93a43febfa732d8b3073d","side":"left"},{"sibling":"eed910228a77ed25a0bdd75b85a1196ad5a57dc8df40499978fe76ba9421a43a","side":"right"},{"sibling":"2c2791320244bc1d327acd0b38fb838ba83b8f5b6baf225017395169cd9b504f","side":"left"},{"sibling":"07df7b4096e4eead45265dbb811864684b1cfdd0b8a72b68a461d2c160055c07","side":"left"},{"sibling":"0fa191199f785899b5cc0f48838a3ffb2aa741ddcd9510fe1664869d36a8d31a","side":"right"},{"sibling":"e6aaada16cbe58b790855c6723700fd07821cc6df59048dc470d404b1ba60842","side":"left"},{"sibling":"d337a9fdcfc121e9691d8db9173af6a3fe0c33d4a6d5f0c8a7a01af04f9fb856","side":"left"},{"sibling":"8b39e07457f5cc5d687d2ae42284dbe705bb87084e7db4b626aff81e51dacd19","side":"right"},{"sibling":"79a713e1e345ccb99c5fe994a11708c8e9bcfa2e91f940d70421cb7d8d77ecc6","side":"right"},{"sibling":"249870fb494bef050c409081e5de45f9042938d7b2823524ee496f296aa63667","side":"right"},{"sibling":"96c48ee8328f1b7925a4cc4421df5cb0bd81c92a5d8354c93126fa5f0166d225","side":"right"},{"sibling":"35ca36cee447f0ef7064a25d55f59357c66901e31427729c4c1d8b14aa8adb6c","side":"left"},{"sibling":"4e13b4a3e69bb83d13913477a782913ce03937edd65046853c1964d3cbb6564b","side":"right"},{"sibling":"0f7b2df1c4580bf7bb7c24b9158ba20593a06af18a5f18d0973e5eff20c35cd8","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":148601,"merkle_root":"44900cffd986f40535c83f46a250e86fbf1019d41f27080a00fbf9b8d77ec33a","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260524T133701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-24T13:37:32Z","sig_algorithm":"ed25519","signature":"059e428c3c5241de303721ad6ac7b748758372180f3f0a717810312aecd6fab073abb3ea264157666de581a2b361c4c6e2aa8ca081b08ce9be090721f0e3400e","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_eb009b4921ebd07c33f6f2b02c7e2e054f39bcab59527b95a9ae2a091c8691a2"}}