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We show that grokking is a norm-driven representational phase transition in regularised training dynamics, and establish the Norm-Separation Delay Law: $T_{\\mathrm{grok}} - T_{\\mathrm{mem}} = \\Theta(\\gamma_{\\mathrm{eff}}^{-1} \\log(\\|\\theta_{\\mathrm{mem}}\\|^2 / \\|\\theta_{\\mathrm{post}}\\|^2))$, where $\\gamma_{\\mathrm{eff}}$ is the optimiser's effective contraction rate ($\\gamma_{\\mathrm{eff}} = \\eta\\lambda$ for SGD, $\\gamma_{\\mathrm{eff}} \\ge \\eta\\lambda$ for AdamW). The upper bound follows from a discrete Lyapunov contraction argument; the matching lower bound from dynamical constraints of regularised first-order optimisation. Across 293 training runs spanning modular addition, modular multiplication, and sparse parity, we confirm","title":"The Norm-Separation Delay Law of Grokking: A First-Principles Theory of Delayed Generalization","url":"https://arxiv.org/abs/2603.13331","vendor":"arxiv_cs_ai"},"summary":"arXiv:2603.13331v2 Announce Type: replace \nAbstract: Grokking -- the sudden generalisation that appears long after a model has perfectly memorised its training data -- has been widely observed but lacks a quantitative theory explaining the length of the delay. We show that grokking is a norm-driven representational phase transition in regularised training dynamics, and establish the Norm-Separation Delay Law: $T_{\\mathrm{grok}} - T_{\\mathrm{mem}} = \\Theta(\\gamma_{\\mathrm{eff}}^{-1} \\log(\\|\\theta_{\\mathrm{mem}}\\|^2 / \\|\\theta_{\\mathrm{post}}\\|^2))$, where $\\gamma_{\\mathrm{eff}}$ is the optimiser's effective contraction rate ($\\gamma_{\\mathrm{eff}} = \\eta\\lambda$ for SGD, $\\gamma_{\\mathrm{eff}} \\ge \\eta\\lambda$ for AdamW). The upper bound follows from a discrete Lyapunov contraction argument; the matching lower bound from dynamical constraints of regularised first-order optimisation. Across 293 training runs spanning modular addition, modular multiplication, and sparse parity, we confirm","title":"The Norm-Separation Delay Law of Grokking: A First-Principles Theory of Delayed Generalization","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-06T04:43:18Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2603.13331"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:0a9833e98d9732a60412e743e9333dae85de38cb826b5e7ced976e4a8a8b4b8461892d51b41c6323fbbaffb9f3806b4cfa6854e70ef501758243f1eb3acbf503","signer":"crovia.substrate","subject":{"observed_at":"2026-05-06T04:43:18Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2603.13331"},"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":"714de1863c17120d81980db1e684cc7141888d15721fcb2a6e42265e83a88db4","leaf_index":116426,"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":"58517d27407bb41b3bc7c90e40269418c1fde84cb60b50ac79461d3af473f9f9","side":"right"},{"sibling":"a96c1b75ca50a3a667956551dab85946499ec5d8cb69c13c0899da2283e89528","side":"left"},{"sibling":"fda2849e27570f38f332e4532b12a23b1eace6ae5702f08b3abeeeafa5bae1a2","side":"right"},{"sibling":"494f8952ec53862535b4710d6e1eb9ee2aa6b0006bfe18d541c52f56700e7745","side":"left"},{"sibling":"e1f695cc914568bfb85a2a8663eb37978d2af9431454081c5a6176747665a0ad","side":"right"},{"sibling":"ef5fe04b64cf0fb50e71dc2d999ac13d32dff59697808d11301613ee8e663a00","side":"right"},{"sibling":"4cc9938ba74578c39e57576728cd5667cffc329ca7f59abc83c2dfe415d03818","side":"left"},{"sibling":"ae2d4d1f9926c807a390f0c20c0a0e8fc89204c3e1f66866d3ea0fda3757402d","side":"left"},{"sibling":"7a43f13f6340a269b939be4b6e73a199a4e8061f0cf9dbb9ba629243174142d3","side":"right"},{"sibling":"c750384c973eaf46aa237cdabe7a75729862dd4101dca10cb8e0626e0ec2d34a","side":"left"},{"sibling":"ddc060ac400459417799f688c75d5b271636afa40d89ec7fc98652473a8061f6","side":"left"},{"sibling":"282afa51266e47629e34d808a360bbb276348bcc5a24bdcc93e83a76e580293e","side":"right"},{"sibling":"05f89b32c00462e60adf95c1fe4579cdc2791b36e8b17573d8f3b5fd5da95a0b","side":"right"},{"sibling":"8ccd9937a2c0d5c04044d07d1557791b7d07bb31eac41a39a675608d44b38f23","side":"right"},{"sibling":"3a5e69cf0803f4c91f3895ed7c9a95748fef240bec4422e167c05300f79f06c0","side":"left"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"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":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","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_5c72068dd56f6f1c1b66992904a2b4f5ec8b8ec04b999ff187aa640e0a9efd1b"}}