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The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose morphisms are *smooth adaptive arrangements*, which consist of a reactive parameter space, a lens given by smooth output and input maps, and a real-valued potential.\n  The main technical result of the paper is what we call *lens internalization*, a lax symmetric monoidal functor Lens(C) $\\to$ C associated to any symmetric monoidal closed category C. Using it, we provide two functors $\\Phi_\\text{phase}$, $\\Phi_\\text{conf}$: Arr $\\to$ PC into the 2-category of polynomial coalgebras -- input-output discrete dynamical systems -- which we take as the semantics category. $\\Phi_\\text{phase}$ stores both position and momentum, whereas $\\Phi_\\text{conf}$ stores only position.\n  When applied to a parameterized fu","title":"Compositional Dynamics in Learning and Mechanics","url":"https://arxiv.org/abs/2606.28984","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.28984v1 Announce Type: cross \nAbstract: We give a single compositional setting in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics. The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose morphisms are *smooth adaptive arrangements*, which consist of a reactive parameter space, a lens given by smooth output and input maps, and a real-valued potential.\n  The main technical result of the paper is what we call *lens internalization*, a lax symmetric monoidal functor Lens(C) $\\to$ C associated to any symmetric monoidal closed category C. Using it, we provide two functors $\\Phi_\\text{phase}$, $\\Phi_\\text{conf}$: Arr $\\to$ PC into the 2-category of polynomial coalgebras -- input-output discrete dynamical systems -- which we take as the semantics category. $\\Phi_\\text{phase}$ stores both position and momentum, whereas $\\Phi_\\text{conf}$ stores only position.\n  When applied to a parameterized fu","title":"Compositional Dynamics in Learning and Mechanics","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-30T04:43:04Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.28984"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:21ff30c7d5506e35d79207b48015eaf80f001cdb0df368611650c11064a6b12a951a3c72192d332c39ceb6e9addee895fb3fb9e497d60c86e07464af21b54e0f","signer":"crovia.substrate","subject":{"observed_at":"2026-06-30T04:43:04Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.28984"},"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":"5a8b41aee497f8983f3939b7f576505e94c48061a7e0c1d3782c6eab53c61772","leaf_index":264844,"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":"b7e72789e5c697fb0162ab7e156418282f4882ab8f1f2d67d4ae18be162815b7","side":"right"},{"sibling":"17db01d3deda2a9c3ddb98e42687d1933c51995b059d9648c2d5bc5c5f6e72e9","side":"right"},{"sibling":"cb835cb53ce4da284aaac8e8454817df4934b0f7f5231705723aec3d01d9d7a0","side":"left"},{"sibling":"3d471b77991037cfcb9f582890653a2137ef8eaa4a7c47b4259ad38dee540066","side":"left"},{"sibling":"445b7076534b1ae8eb580e2a117d3733164a69e9542c90df52250a3879236f9e","side":"right"},{"sibling":"21aeb247f596e755f92e03c02397140679f193fb3b000cc11078505b42970abc","side":"right"},{"sibling":"07528e32fda4a58ff6ab8ca2268cc78d4ff4ed66bfadde9432f593a2fb4e74e7","side":"right"},{"sibling":"86ba07f5dfcbf22e044b5aa0ecc15cd2f29de9218e54da603c4ffe753c522516","side":"left"},{"sibling":"5767f14b55df4212d25ae2f52b8f3d4abff2555c65ba62935ed6cb0a8e617b43","side":"right"},{"sibling":"1549a8883ab3267f958dc2624919e40c65c82958b8005967ed6a4da1247da0ad","side":"left"},{"sibling":"23994bf0974e5c9c7f63a61b4f0a48b0ca756a4adc34a8f85f878e774c37dfbe","side":"right"},{"sibling":"f9b4bed84fa6990c71ad2887c91bda183001f05f6b648f21d1273045a26b11fd","side":"left"},{"sibling":"173d2dc4b29ee04ea41d6d0ebc334c4bc2d46e7ee4230c94765413f24fb4bc42","side":"right"},{"sibling":"112461f7c0ec411116fb5c6c90fe95cea9d8f188b9fe08afe25a837ac02d0071","side":"right"},{"sibling":"ea9488204352c49db8f7daf05eefcd7628ecf9413830346674801a99d0654a94","side":"right"},{"sibling":"6261c13b9922cb657f10d1e5d36ec15d8771cf8766e36c61dcbffb7bed57e396","side":"right"},{"sibling":"fa19aa3faf287618b820bcfceebb366152ad521dd20ef9f51e977816663e448b","side":"right"},{"sibling":"c32f943406b62d1fc59b7f7e243492174c8e1caba8c8a2705f86c773315736e0","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":265374,"merkle_root":"9636001ecab173cb6af10dc7c71eb14585daa62f9c0a6f027046f05633156891","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260630T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-30T05:38:03Z","sig_algorithm":"ed25519","signature":"6d4b8fd9b9da856cbb5fba7540877c6a63fa18a5ec3eaf28dc4d6d1c64921c9c0565f8c95f4b7aec0e7744fd7754844f051baf863db708cad768765b416a7b0c","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_fd20e2ff74aa6629e5fa6a82972a337f898e8ef9ee413b3c089c6a651f3245b7"}}