{"_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_24012c1f08976a888070a0b54c349c71a3858e30b17ebc37a9b9f37bf4857ad2","bitcoin_anchor":{"bitcoin_attestations":["bitcoin_block_949451"],"calendar_attestations":["https://finney.calendar.eternitywall.com","https://btc.calendar.catallaxy.com","https://alice.btc.calendar.opentimestamps.org","https://bob.btc.calendar.opentimestamps.org"],"ots_url":"/registry/data/substrate/anchors/77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3.ots","stamped_at":"2026-05-15T03:00:03Z","status":"bitcoin"},"envelope":{"anchors":[{"chain":"crovia.axiom_graph","height":0,"merkle_proof":"spider_arxiv_retraction_v1","root_at_anchor":"spider_arxiv_retraction_v1"}],"axiom_id":"axm_24012c1f08976a888070a0b54c349c71a3858e30b17ebc37a9b9f37bf4857ad2","axiom_type":"AX.OBS","body":{"axiom_subtype":"research.arxiv_retraction.v1","category":"research","fingerprint":"cbcaeb42a14fb1e6ed581c4098439687003797234eb44462b6b58bf018fe2d42","published":"2026-05-04T07:11:11Z","receipt_hash":"cbcaeb42a14fb1e6ed581c4098439687003797234eb44462b6b58bf018fe2d42","schema":"spider.research.arxiv_retraction.v1","spider":"arxiv_retraction","spider_record":{"axiom_subtype":"research.arxiv_retraction.v1","category":"research","decision_hint":"POSITIVE","envelope_target":"AX.OBS","fingerprint":"cbcaeb42a14fb1e6ed581c4098439687003797234eb44462b6b58bf018fe2d42","observed_at":"2026-05-05T05:55:17.011995Z","parent_run_hash":"a87be6250df7e7a8dd886b916f53fee983afed6ab5b98a08bad61550cbe93026","published":"2026-05-04T07:11:11Z","runtime_version":"0.1.0","schema":"spider.research.arxiv_retraction.v1","source_status":200,"source_url":"http://export.arxiv.org/api/query?search_query=cat:cs.LG+AND+%28abs:withdrawn+OR+abs:retracted%29&max_results=20&sortBy=submittedDate&sortOrder=descending","spider":"arxiv_retraction","summary_excerpt":"Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing literature often presents results at a high level of abstraction, omitting the detailed coordinate-level derivations required for implementation and algorithm development.\n  This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry, with a focus on explicit derivations tailored to Riemannian optimization. We systematically develop the key geometric structures -- including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics -- emphasizing step-by-step derivations in coordinates and matrix form.\n  Building on these foundations, we derive the Riemannian gradient, Hessian, exponential map, and retraction in a form suitable for numerical computation. We furt","title":"Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations","url":"https://arxiv.org/pdf/2605.02279v1","vendor":"arxiv"},"summary":"Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing literature often presents results at a high level of abstraction, omitting the detailed coordinate-level derivations required for implementation and algorithm development.\n  This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry, with a focus on explicit derivations tailored to Riemannian optimization. We systematically develop the key geometric structures -- including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics -- emphasizing step-by-step derivations in coordinates and matrix form.\n  Building on these foundations, we derive the Riemannian gradient, Hessian, exponential map, and retraction in a form suitable for numerical computation. We furt","title":"Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations","vendor":"arxiv"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-05T05:55:17Z","notes":"Spider arxiv_retraction (research) research.arxiv_retraction.v1","object":{"captured_by":"crovia.spider.arxiv_retraction","primary_source_url":"https://arxiv.org/pdf/2605.02279v1"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:2ac2bfc7a6c6ea5908a1ddc29ac313d3b6d34247006ea1865acacc195c3d6394c8fe2a208f5803fd69cf4394cfa79076170ae3df63a59d38d804a80ef4e98409","signer":"crovia.substrate","subject":{"observed_at":"2026-05-05T05:55:17Z","source_collector":"spider:arxiv_retraction","target_id":"https://arxiv.org/pdf/2605.02279v1"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/research/arxiv_retraction_v1.jsonl\",\"source_seal_merkle_root\":\"spider_arxiv_retraction_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"dc5920692e72eec1b8ebdc4ac41dd9400432dc1154c9bf365ab3e39fad241ae9","leaf_index":114618,"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":"91a1ffb1f8d99608a0f53416abbace62ade6642e55e51ab231135e1e2377277a","side":"right"},{"sibling":"f5088c69274772db52a8cb13997f34597d4088abf511370d0c262a896e9d1f44","side":"left"},{"sibling":"8c4dd281327aa4b5d02f9b0230bf8777ae2cfe989010c2c486abe8e4f4f7bfc8","side":"right"},{"sibling":"719041e71811b14458ac65cb4ebc6af10bac9ebb72cbf82a39bd52a685326fb8","side":"left"},{"sibling":"84c08c3560c13694a876fa3a6a475956fb902442fdb265b5329a7b2de7ce8760","side":"left"},{"sibling":"d86313f94e2c873d65940aa9ae50cb44fc5e8664802bc290cf5770cad94574a9","side":"left"},{"sibling":"348269778c40afb3531ae3935f9602f8db1294ab94e396ac4ce0ff2654852549","side":"right"},{"sibling":"2ca18b21a5a7252cca79c594b603a614a8053a95d7710dcefe3a448017e4ef70","side":"left"},{"sibling":"79216ea1cdb095fddf30c2914a5fd7ccfe00cef50c3af9b506288bdce49818ef","side":"left"},{"sibling":"8df6c69f776fbba620634f6baf125eeaa6fdf9426e461e649028010d9a703f42","side":"left"},{"sibling":"ea40d1bd0432ad4dd90f023c692aaab8c8f54e27ce652e82f2ec8e16cacfb632","side":"left"},{"sibling":"673cd6d27b696232f7f65e7a0733c7df6bc9fac3bcb569907ebb0de1c72c4bb2","side":"left"},{"sibling":"76d7c4385d71ae8104107105be66eddb792e01ed6e493db5a9b4eec5441abb76","side":"left"},{"sibling":"b03f005860bf95148ea89c703a32ca19ab7a2ca71b58bb628ff94a9edb8703b0","side":"left"},{"sibling":"21d1e30b556f553dc939fddc23e6367f0a7755ebe3dd489dc0001cef017beb7f","side":"right"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"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":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","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_24012c1f08976a888070a0b54c349c71a3858e30b17ebc37a9b9f37bf4857ad2"}}