{"_canonicalization":{"envelope_id":"axm_ + sha256(envelope minus {signature, axiom_id, anchors})","envelope_signature":"ed25519(envelope minus {signature, axiom_id})","json":"sort_keys=True, separators=(',',':'), ensure_ascii=False, allow_nan=False, utf-8","leaf_hash":"sha256(0x00 || canonical_json(envelope_full))","seal_signature":"ed25519(seal minus {signature, sig_algorithm})"},"axiom_id":"axm_deecfeff3f9917d6bf502824dcb7f116a0f8d4d57ea0298b350a97df24c03de8","bitcoin_anchor":{"bitcoin_attestations":[],"calendar_attestations":[],"ots_url":"","stamped_at":"","status":"pending_next_stamp"},"envelope":{"anchors":[{"chain":"crovia.axiom_graph","height":0,"merkle_proof":"spider_vendor_press_v1","root_at_anchor":"spider_vendor_press_v1"}],"axiom_id":"axm_deecfeff3f9917d6bf502824dcb7f116a0f8d4d57ea0298b350a97df24c03de8","axiom_type":"AX.OBS","body":{"axiom_subtype":"news.vendor_press.v1","category":"news","fingerprint":"59d092c86a6cbf5f00ef8a3a34d52daca112b7fcdca39cf26a8f1f463183c029","published":"Tue, 28 Jul 2026 00:00:00 -0400","receipt_hash":"59d092c86a6cbf5f00ef8a3a34d52daca112b7fcdca39cf26a8f1f463183c029","schema":"spider.news.vendor_press.v1","spider":"vendor_press","spider_record":{"axiom_subtype":"news.vendor_press.v1","category":"news","decision_hint":"POSITIVE","envelope_target":"AX.OBS","fingerprint":"59d092c86a6cbf5f00ef8a3a34d52daca112b7fcdca39cf26a8f1f463183c029","observed_at":"2026-07-28T04:43:08.282317Z","parent_run_hash":"23a1ef85134515049ced29518443d084afc46fd7c967741e6c6acdbdbbf29939","published":"Tue, 28 Jul 2026 00:00:00 -0400","runtime_version":"0.1.0","schema":"spider.news.vendor_press.v1","source_status":200,"source_url":"https://export.arxiv.org/rss/cs.AI","spider":"vendor_press","summary_excerpt":"arXiv:2607.23967v1 Announce Type: new \nAbstract: Delayed generalization, or grokking, remains poorly understood despite extensive empirical study. We identify an exactly solvable late-time relaxation mechanism for grokking in linear models trained with full-batch heavy-ball optimization and weight decay, together with a locally quadratic extension to nonlinear neural networks. Our analysis reveals a distinguished population-active component of the empirical null space, which we call the grokking subspace. Along this subspace, the training predictions remain unchanged, leaving weight decay as the sole restoring force and giving rise to a slow dissipative relaxation governed by an exact discrete-time and continuous-time law. We show that only this subspace contributes to the slow asymptotic decay of the population risk and derive explicit iteration-scale predictions for the grokking time, recovering the familiar $(1-\\beta)/(\\eta\\lambda)$ scaling in the weak-regularization regime. The the","title":"Grokking on the Weight-Decay Clock: A Rate Hierarchy from Softly Broken Symmetries","url":"https://arxiv.org/abs/2607.23967","vendor":"arxiv_cs_ai"},"summary":"arXiv:2607.23967v1 Announce Type: new \nAbstract: Delayed generalization, or grokking, remains poorly understood despite extensive empirical study. We identify an exactly solvable late-time relaxation mechanism for grokking in linear models trained with full-batch heavy-ball optimization and weight decay, together with a locally quadratic extension to nonlinear neural networks. Our analysis reveals a distinguished population-active component of the empirical null space, which we call the grokking subspace. Along this subspace, the training predictions remain unchanged, leaving weight decay as the sole restoring force and giving rise to a slow dissipative relaxation governed by an exact discrete-time and continuous-time law. We show that only this subspace contributes to the slow asymptotic decay of the population risk and derive explicit iteration-scale predictions for the grokking time, recovering the familiar $(1-\\beta)/(\\eta\\lambda)$ scaling in the weak-regularization regime. The the","title":"Grokking on the Weight-Decay Clock: A Rate Hierarchy from Softly Broken Symmetries","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-07-28T04:43:08Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2607.23967"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:6c01bf56f3db14b60b82de022035a1f27577ab7b7bb795aaf1f2dd50cb05a9c58b7feb19e5b5ef46dd22895d924e25dc6ceb140793f033e8029ec7eb70f4e108","signer":"crovia.substrate","subject":{"observed_at":"2026-07-28T04:43:08Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2607.23967"},"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":"22b1ff5e8bd12acd06e16e368a3b729e9716d8aeda911de0dff4cf76f7583211","leaf_index":360451,"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":"f34432fb40183aaa79c466a85e1f19feeff8a197a5fae642b097d556da662414","side":"left"},{"sibling":"2605ee5a2f354d2578b35e7aff61008d13e8d57fd4f37f867395e2ae239de718","side":"left"},{"sibling":"333af1692139b9d146a7e7037741af6be78100dda1697a49715a9961b9f2af71","side":"right"},{"sibling":"b0b969f137beb704c98eeb7d1c91486270fc3f2e8824fbd4142d0ed51fe60293","side":"right"},{"sibling":"75f00d814548b9c728ef449fab945f4caa696d0a5c900c84208ce67d54f1cba5","side":"right"},{"sibling":"7a6d1913b8fd5eb095eeedf6ef159b2ad2fda0579ffedbad4c51f4c30689a4df","side":"right"},{"sibling":"af35a1d32b2197ab764f9882c8ba5ae653d1e5419b2df50a1daf88021ca8b2bd","side":"right"},{"sibling":"7da15787b28ee42d6f6ffa689ccb25f3c7958b4ad112266a2a4acad4347e0386","side":"right"},{"sibling":"eb79f1d599c07786d2268140481af8b617999ecc5688c6283fe58e5e6f3a2640","side":"right"},{"sibling":"fe03c6d0b083c4097049d7fc6d7d08dfdb05d9c198b2786336689712d66eabf0","side":"right"},{"sibling":"590ea76fdfc1b9e8072055c378be3182f916709e8d06bd955232e650a7188b86","side":"right"},{"sibling":"d92781c59301ffd5bfb0bad75d9fdf6d73879f715518149d383f7362213e0daf","side":"right"},{"sibling":"f315303d4402b57497416d48eb4c4bb50405b40862d41c7caf318cb3d29c5237","side":"right"},{"sibling":"36973eb5f586cd67e0c0dc055dd87e734aa544c35d4400d2c9f932f8ef8fb27f","side":"right"},{"sibling":"e39f7900355489c4718b21cc2d3d06382e1d2f22b864d10be9ebc9d498b279a1","side":"right"},{"sibling":"1f9a970b25dd938c98cabc9e0a55c5a6f46292b9fa37cc89d90ef0cbb1e05a8c","side":"left"},{"sibling":"77025bcb374a7ad74f520643e20a8ae1205a7ee78507b0beb93117760f1c29d3","side":"left"},{"sibling":"3b50864499c874394ea0928567747666eaf59b01380e46cd52164ec5acec0f71","side":"right"},{"sibling":"1cecb7f447febd025aac272837c80de218aecc6485d2395a509b2a1f1b9c746e","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":361008,"merkle_root":"3065e8369ea437c06beba806dc4e4bb159979adeb21fe632242c1906a7204647","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260728T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-07-28T05:38:48Z","sig_algorithm":"ed25519","signature":"9141644407577a82611c1579110f667de2d46dc6b93c6322edf26f4c3056ea99f0e56502853908e30d87c38bcf95eb6e0ab5130525aa51505bd6f61938120609","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_deecfeff3f9917d6bf502824dcb7f116a0f8d4d57ea0298b350a97df24c03de8"}}