{"_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_6c55b35d16f3c5c9d12db2cb753069e58e1588291cbb9b4389f8a8e74996713b","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_6c55b35d16f3c5c9d12db2cb753069e58e1588291cbb9b4389f8a8e74996713b","axiom_type":"AX.OBS","body":{"axiom_subtype":"news.vendor_press.v1","category":"news","fingerprint":"df283b62fa1dd2782ed6c12784af086140036b43215279459d4e4db58a433003","published":"Tue, 09 Jun 2026 00:00:00 -0400","receipt_hash":"df283b62fa1dd2782ed6c12784af086140036b43215279459d4e4db58a433003","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":"df283b62fa1dd2782ed6c12784af086140036b43215279459d4e4db58a433003","observed_at":"2026-06-09T04:43:45.619596Z","parent_run_hash":"f2344865fd128464efd1bacba326b5a7ccea707694b8c5650dd51ae8c46ac8a1","published":"Tue, 09 Jun 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:2606.07600v1 Announce Type: cross \nAbstract: We formulate data propagation through the Transformer, the machine learning architecture powering large language models, as a nonlinear control system on the space of probability measures. For the mean-field Transformer model with self-attention and affine feed-forward layers, we prove that Gaussian distributions remain exactly Gaussian along the induced flow. This invariance reduces the infinite-dimensional measure dynamics to a finite-dimensional bilinear control system governing the evolution of the mean and covariance, reformulates the expressive capacity of Transformers as a reachability problem for prescribed Gaussian moments, and reveals a novel connection with Riccati-type equations from classical filtering and control.\n  For time-varying controls, we prove exact finite-time reachability of any target Gaussian distribution whose covariance matrix has the same rank as the initial one, this rank constraint being an intrinsic inva","title":"Reachability and asymptotics of Gaussian Transformer dynamics","url":"https://arxiv.org/abs/2606.07600","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.07600v1 Announce Type: cross \nAbstract: We formulate data propagation through the Transformer, the machine learning architecture powering large language models, as a nonlinear control system on the space of probability measures. For the mean-field Transformer model with self-attention and affine feed-forward layers, we prove that Gaussian distributions remain exactly Gaussian along the induced flow. This invariance reduces the infinite-dimensional measure dynamics to a finite-dimensional bilinear control system governing the evolution of the mean and covariance, reformulates the expressive capacity of Transformers as a reachability problem for prescribed Gaussian moments, and reveals a novel connection with Riccati-type equations from classical filtering and control.\n  For time-varying controls, we prove exact finite-time reachability of any target Gaussian distribution whose covariance matrix has the same rank as the initial one, this rank constraint being an intrinsic inva","title":"Reachability and asymptotics of Gaussian Transformer dynamics","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-09T04:43:45Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.07600"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:fd6a2a514fedfcbeb5f5a07c414afbabbc719cc0d1e75d045a460053f5be2e181a6257c83e499d23e29ec330a71f8163dd8a15046c198ce4598313cb9ac1da0c","signer":"crovia.substrate","subject":{"observed_at":"2026-06-09T04:43:45Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.07600"},"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":"c8462b41ae00302e28eef08cb90a8360f2538b8eb88cb6fa3517960864807a3d","leaf_index":224128,"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":"a1762daea3c3062e977f5f32fc2077029515f2e28596349177b530149535acae","side":"right"},{"sibling":"15f413e06b3596f9ea81f9eb79ec083fb259463aba9076fb8ae341e1be739549","side":"right"},{"sibling":"e392baa763725938b39a337f09fd159ffcb546bde22725a7e0893cb686307c57","side":"right"},{"sibling":"3a2241ca1f64d9cbaf511aa86b89a0e443dc49d25e47e12be3e599a664254eeb","side":"right"},{"sibling":"a9253ba65ae4c18e6e33d131b3270e612f45ffc53416f1ec8315760284766078","side":"right"},{"sibling":"a915a185e4c11882e44a7c9abc37c2b7d1966a55d1732b8518c0b0d92f8f8cec","side":"right"},{"sibling":"0be0c215d1fb2f707cd98f8ae4ceb30761bee17955a0bb50300a2e9fe5db1e0e","side":"right"},{"sibling":"10fa631530dc46d85fa0d114403a05652fa3690c3219b57f26650e0db77a8036","side":"left"},{"sibling":"dd5d61819b080d26c4380e26887a8eb889d51e2c022f796a39855defe6f68d95","side":"left"},{"sibling":"e8dcea313a54920d83e4f72d5a4223f986c719f264241d171c4712efaaf1fc54","side":"left"},{"sibling":"5480e1ea31f4744f9bd7c4261771fe51f2cdb01e705cc17320bfc202d935ca12","side":"right"},{"sibling":"24fdc29d461691aedb6fa920758206b5bb43851f477ef7a04c34aaed84b8971b","side":"left"},{"sibling":"036922da4e1e2c46d948f070454bfad299b7406fb00735ea9d8bd1e687f5f445","side":"right"},{"sibling":"533d82482604463aa4a281b9d8b85917b383b7c5f924b2b494039524c55e8797","side":"left"},{"sibling":"b2590791b920ca2a4ed39de126d2c0b1a10d9e7e62f572f12425f214e767b6e1","side":"left"},{"sibling":"87c6b850dfec08ac35a693d9db3a3315250a68adb1cfab9b1015f212b63b15bd","side":"right"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"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":224761,"merkle_root":"e9f7b49b652e869ab97ffba9c5a31356b2d0e3dc5d00bb28944adf737c46b1e7","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260609T103805Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-09T14:15:34Z","sig_algorithm":"ed25519","signature":"8ad8076fb12c8e486ae1d1559a9a7ba8e2ee996a9ad3d8ba7bcdbdbd88ab3a15bcb429707aca6d3e9d8b97e2ba755b3dcc77b1abb6601ccb829842719a6fb30d","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_6c55b35d16f3c5c9d12db2cb753069e58e1588291cbb9b4389f8a8e74996713b"}}