{"_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_5295a64fff469ea35c5656222569796b2773faf5da99fdfa93be72da08cf9db1","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_5295a64fff469ea35c5656222569796b2773faf5da99fdfa93be72da08cf9db1","axiom_type":"AX.OBS","body":{"axiom_subtype":"news.vendor_press.v1","category":"news","fingerprint":"853eed847815d4d00a57aba5598c94227a057db2f72eb75408c4f9c09947ac95","published":"Fri, 15 May 2026 00:00:00 -0400","receipt_hash":"853eed847815d4d00a57aba5598c94227a057db2f72eb75408c4f9c09947ac95","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":"853eed847815d4d00a57aba5598c94227a057db2f72eb75408c4f9c09947ac95","observed_at":"2026-05-15T04:43:17.611638Z","parent_run_hash":"5c64f85625fabd323e9c4a1cf068c012fb88a248deda9a9ac702fb2f9799f2e5","published":"Fri, 15 May 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:2602.19533v2 Announce Type: replace-cross \nAbstract: This paper investigates the grokking phenomenon, which refers to the sudden transition from a long memorization to generalization observed during neural networks training, in the context of learning multiplication in finite-dimensional algebras (FDA). While prior work on grokking has focused mainly on group operations, we extend the analysis to more general algebraic structures, including non-associative, non-commutative, and non-unital algebras. We show that learning group operations is a special case of learning FDA, and that learning multiplication in FDA amounts to learning a bilinear product specified by the algebra's structure tensor. For algebras over the reals, we connect the learning problem to matrix factorization with an implicit low-rank bias, and for algebras over finite fields, we show that grokking emerges naturally as models must learn discrete representations of algebraic elements. This leads us to experimental","title":"Grokking Finite-Dimensional Algebra","url":"https://arxiv.org/abs/2602.19533","vendor":"arxiv_cs_ai"},"summary":"arXiv:2602.19533v2 Announce Type: replace-cross \nAbstract: This paper investigates the grokking phenomenon, which refers to the sudden transition from a long memorization to generalization observed during neural networks training, in the context of learning multiplication in finite-dimensional algebras (FDA). While prior work on grokking has focused mainly on group operations, we extend the analysis to more general algebraic structures, including non-associative, non-commutative, and non-unital algebras. We show that learning group operations is a special case of learning FDA, and that learning multiplication in FDA amounts to learning a bilinear product specified by the algebra's structure tensor. For algebras over the reals, we connect the learning problem to matrix factorization with an implicit low-rank bias, and for algebras over finite fields, we show that grokking emerges naturally as models must learn discrete representations of algebraic elements. This leads us to experimental","title":"Grokking Finite-Dimensional Algebra","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-15T04:43:17Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2602.19533"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:94f2ffb7f4876e8f48d1380d15cf2cbf2e0eae24114279a1bca3c2a84b7f1777559fef4f8d131fc2baaf9d8fa2f5dbf6eb1aaab6ab37d9f5089e7ebacff8820e","signer":"crovia.substrate","subject":{"observed_at":"2026-05-15T04:43:17Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2602.19533"},"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":"7d08c64af0b4e4286b0b422ba55ec4f70d13b064e9bb579e98e57fcdee085b22","leaf_index":134775,"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":"61e6c28826a1d5128eaeec8c0f8b762cc38444de6ec2038f5f6ffa72c36ec1fb","side":"left"},{"sibling":"5d4a6e16da6c5309e9aa98015eb754b736941af6f575ab40195b95e164bee386","side":"left"},{"sibling":"8e37515c762220c29ccd140eee7acdd1233890c8ed6a1190a11ee4ef4fea422d","side":"left"},{"sibling":"cfe1fbbaf9ee8a3e99a75f4253419fe01a772df4cbee2f354b2093881ceeb077","side":"right"},{"sibling":"c36d1229c95a999aa6441135705fd15d2b30eca6083ce6b83f5a334666414b5c","side":"left"},{"sibling":"952bb2ea9b8e1bb4f01c53d0e4d6522140e49683ac624344442d5a1522d7c3be","side":"left"},{"sibling":"4999a953ceed25471a2bccd6eadce4cee20773995bf2d506b76320a8705da9a4","side":"left"},{"sibling":"d8ac22bfe982d5dc7f00879b0b1dfae8f4e95a38a6242933cbebb3b7271d5704","side":"right"},{"sibling":"a17a0eaa87574a308e2c02cf125072b9e69566b9e8d6d2e64a02880192b885d5","side":"right"},{"sibling":"6bd0475fd3a73322a4f73095c96b072de89e462ce5839161aa552e0208619bce","side":"left"},{"sibling":"36672459e5ed50c64ee1842b69cb6d2eb682c2a04844555be8d124257571994a","side":"left"},{"sibling":"727783827652adfa99c455bd80a01bfb33836228e51068b4f654ef3da468ca69","side":"left"},{"sibling":"623194cd30880ed223e306737fdb111aa0d781751bfc47553c404a6af6aad2c4","side":"right"},{"sibling":"fc8f53ed42756907fb79ee19a4ed09f72c560e5302b3d98198b96bf1da635a4a","side":"right"},{"sibling":"d6607539da7ba39ec68be2d12f27ed6768766c745e3120fd915f88c5e288e07c","side":"right"},{"sibling":"b63408a424d27cd6a75e0fb155e69a58a328f41e9cb9dba1eddef9a5289cc7fd","side":"right"},{"sibling":"356fb36a4e188f03d7a05c54cd8789bdd40eda454b9bc9560f667acc08e6c4e0","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":134886,"merkle_root":"c6c7ae28c065bced89e7f844216b073f1a7cc4b378db0d41a98bcd21b28066db","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T05:37:26Z","sig_algorithm":"ed25519","signature":"5a3978c26017daf4104adbb3e1c3099c5750157acbfc7242ece1815dc6740fe08a690291ce0afe42011e20cc565b5ebe64ec016bf658b5bab563a63337985c05","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_5295a64fff469ea35c5656222569796b2773faf5da99fdfa93be72da08cf9db1"}}