{"_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_cecd3753b56c0aac15bed3115c07fbaba0db7bb7301c2c548afab4e4e22b0efd","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_cecd3753b56c0aac15bed3115c07fbaba0db7bb7301c2c548afab4e4e22b0efd","axiom_type":"AX.OBS","body":{"axiom_subtype":"news.vendor_press.v1","category":"news","fingerprint":"c55bba8964088fe07871af94fdefdaee9fa8c83716e99ba707a76ca86658eaba","published":"Fri, 22 May 2026 00:00:00 -0400","receipt_hash":"c55bba8964088fe07871af94fdefdaee9fa8c83716e99ba707a76ca86658eaba","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":"c55bba8964088fe07871af94fdefdaee9fa8c83716e99ba707a76ca86658eaba","observed_at":"2026-05-22T04:43:12.593252Z","parent_run_hash":"dd4d56660d55b2d65dc84dc5e7c8f83487d90da2dd343b5707d6948a3bb0d917","published":"Fri, 22 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:2605.20440v1 Announce Type: cross \nAbstract: We introduce the $\\star_G$ tensor algebra, in which any finite group $G$ defines the multiplication rule, making equivariance an intrinsic algebraic property rather than an architectural constraint. The framework rests on three machine-verified theoretical pillars: (i)~an Eckart-Young optimality guarantee for the $\\star_G$-SVD: the first such result for symmetry-preserving tensor approximation, exact and polynomial-time; (ii)~a Kronecker factorization that composes multiple symmetries by replacing $F_G$ with $F_{G_1} \\otimes F_{G_2}$ with no architectural redesign; and (iii)~a 600-line Lean~4 formalization of the $\\star_G$ algebra. The framework provides capabilities that equivariant neural networks (ENNs) structurally cannot: a closed-form per-irreducible-representation decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset. As a non-trivial empirical demonstration, decomposing QM9 ","title":"Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery","url":"https://arxiv.org/abs/2605.20440","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.20440v1 Announce Type: cross \nAbstract: We introduce the $\\star_G$ tensor algebra, in which any finite group $G$ defines the multiplication rule, making equivariance an intrinsic algebraic property rather than an architectural constraint. The framework rests on three machine-verified theoretical pillars: (i)~an Eckart-Young optimality guarantee for the $\\star_G$-SVD: the first such result for symmetry-preserving tensor approximation, exact and polynomial-time; (ii)~a Kronecker factorization that composes multiple symmetries by replacing $F_G$ with $F_{G_1} \\otimes F_{G_2}$ with no architectural redesign; and (iii)~a 600-line Lean~4 formalization of the $\\star_G$ algebra. The framework provides capabilities that equivariant neural networks (ENNs) structurally cannot: a closed-form per-irreducible-representation decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset. As a non-trivial empirical demonstration, decomposing QM9 ","title":"Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery","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/2605.20440"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:82e35a5652488f56efe9f4b1a39449481bc504412038e0db223054e10a35eb39d2cd44456999200b75ec959c13aec3472c55b636a96831dd97ca1e20516d4105","signer":"crovia.substrate","subject":{"observed_at":"2026-05-22T04:43:12Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.20440"},"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":"8ab0feeb951de0d1e7033de11ad95411c37885bdc98f6e93b24be5d45ef99514","leaf_index":148225,"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":"f502e6d2d94ab2f2d64dcdc96a673aa68179c7020a8266737a7298b986f9004a","side":"left"},{"sibling":"4eb3dc90690b832eb5e2cd45545ae388a10d787dd0bf9b65ca2cd3f88292ce64","side":"right"},{"sibling":"79c7b1d262fc0490e56c81573d4ea23a324671c069e68a9f660def56087f87a5","side":"right"},{"sibling":"0af8501851a7ffee74657e8b9dcbf1cdf54ef84af8ca15de20634bf6ce588d85","side":"right"},{"sibling":"4719d72db3535b04c69a3b1a92eaf8b76e4ae41300132f462730e0f36d72e2bf","side":"right"},{"sibling":"0cdd6f74518a78e2d51c6d1c6d14d0688bc4da61583f7df49842f7924f36305e","side":"right"},{"sibling":"2f67ce1ca4d405b588b0534baed360bc76fbba6b3678854914e0889a47207afb","side":"right"},{"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_cecd3753b56c0aac15bed3115c07fbaba0db7bb7301c2c548afab4e4e22b0efd"}}