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We develop a kinetic framework for algorithms that estimate both gradient and Hessian from black-box function values. The naive random-direction Hessian estimator turns out to be biased even on quadratics; a Gaussian--Stein correction is needed to estimate the Hessian of the Gaussian-smoothed objective. Linearizing the inverse Hessian exposes two noise channels: gradient noise preconditioned by the inverse Hessian, and Hessian noise transmitted through an inverse-Hessian sandwich. Under a noisy oracle the second channel carries the second-difference factor $\\mu_H^{-4}$. A small-mass kinetic lift links the finite-step Newton update to an underdamped phase-space model; the overdamped spatial limit yields a Lyapunov bound that exposes the curvature--variance trade-off between st","title":"A Formal Kinetic Theory for Zeroth-Order Newton Dynamics:Stein-Corrected Hessian Estimation and Curvature--Variance Trade-offs","url":"https://arxiv.org/abs/2607.22567","vendor":"arxiv_cs_ai"},"summary":"arXiv:2607.22567v1 Announce Type: cross \nAbstract: Zeroth-order Newton-type methods are useful when gradients and Hessians are unavailable, but they behave quite differently from first-order gradient-free methods. We develop a kinetic framework for algorithms that estimate both gradient and Hessian from black-box function values. The naive random-direction Hessian estimator turns out to be biased even on quadratics; a Gaussian--Stein correction is needed to estimate the Hessian of the Gaussian-smoothed objective. Linearizing the inverse Hessian exposes two noise channels: gradient noise preconditioned by the inverse Hessian, and Hessian noise transmitted through an inverse-Hessian sandwich. Under a noisy oracle the second channel carries the second-difference factor $\\mu_H^{-4}$. A small-mass kinetic lift links the finite-step Newton update to an underdamped phase-space model; the overdamped spatial limit yields a Lyapunov bound that exposes the curvature--variance trade-off between st","title":"A Formal Kinetic Theory for Zeroth-Order Newton Dynamics:Stein-Corrected Hessian Estimation and Curvature--Variance Trade-offs","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/2607.22567"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:649461f1f8acaf47f44564706985fd65869968a0af77964631f1fa493059019861477808a0f03c3d566106048f31607b98e8cbb4f32544ea6a144f6e9fcd9a0b","signer":"crovia.substrate","subject":{"observed_at":"2026-07-28T04:43:08Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2607.22567"},"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":"681125cd01d07bfb7940e4b1d500c404fb4e948071b8d0828c6a5173b9cbdacf","leaf_index":360497,"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":"e6022c4b27c0e9d812be462ef37c825ab4e6747b7d5fabf38b57f8ae3096c0c3","side":"left"},{"sibling":"fee184e74ce2dfa7f20322108cc10211c098e7eaf576fe3f6526d1a32385376b","side":"right"},{"sibling":"f208c69496d70c17481bf5d8f4ba3cbe25a73a6752ccc209a5667abfac789fdc","side":"right"},{"sibling":"24daf7e5ef0776e915f14c651a938f7c5a56bc8b0ffcab2749194ab8fac99b91","side":"right"},{"sibling":"cf202c136ceca8b1c7545c1a81da720e5ad0c04565921cc649b297a30a4e6c74","side":"left"},{"sibling":"9a45174405abad8c6238b5cea175009c5728635224ac72d0182f5d73903f2e17","side":"left"},{"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_5f15c8fd708d65e1bbe10b997f3f95e85b14eafb2ccac4ad5cde9f095797dd04"}}