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The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$.\n  Under a bounded local geometry assumption -- bounded cell size $v_{\\max}$, bounded stalk dimension $d$, and bounded nerve degree $D$ -- each edit (vertex insertion, edge insertion, restriction map update) affects only a bounded set of local coboundary blocks. The algorithm therefore processes lazy streaming edits in $O(1)$ time with respect to the total complex size $n$ (with cost polynomial in the local geometry parameters $v_{\\max}$, $d$, and $D$, which are treated as constants independent of $n$), deferring loca","title":"Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry","url":"https://arxiv.org/abs/2606.04227","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.04227v1 Announce Type: cross \nAbstract: We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \\mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$.\n  Under a bounded local geometry assumption -- bounded cell size $v_{\\max}$, bounded stalk dimension $d$, and bounded nerve degree $D$ -- each edit (vertex insertion, edge insertion, restriction map update) affects only a bounded set of local coboundary blocks. The algorithm therefore processes lazy streaming edits in $O(1)$ time with respect to the total complex size $n$ (with cost polynomial in the local geometry parameters $v_{\\max}$, $d$, and $D$, which are treated as constants independent of $n$), deferring loca","title":"Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-04T04:43:08Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.04227"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:adea54651c7df000b1066db675e2c84d9dfb4ec3b4170ba6257c653057f329e703dc374bd0d8dda6adc724dcbf86101c460e7f05aa7d5108c2b21a28eb25840d","signer":"crovia.substrate","subject":{"observed_at":"2026-06-04T04:43:08Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.04227"},"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":"e7684618f9772e808fbe6243b2feabfc6f085108d51d6c65d7e18986d40ae145","leaf_index":212650,"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":"a3b6bc987a2ec1346c17d03c2d8d52281570ca7d099d6aed2b8b27a07d0ca7c3","side":"right"},{"sibling":"6740835c2d216014bc43b8ee66230dbb3e2061754ec06c5edd29f3532ea59588","side":"left"},{"sibling":"fa511d5c15aa498190f1af92bff107477380f24feb36c1344c1b4d769614bf2a","side":"right"},{"sibling":"21fb323395ab525f44b2019cd4b31a8b6b30bce588e56495226ff9b087096196","side":"left"},{"sibling":"7fcfb0763d8a2c325af279a923e44ed6aa720660e48cb87bc03faad86e3e94d3","side":"right"},{"sibling":"481b11a7cac05112b639ea1df21e19d7596edbdc2ad116d4da7aba2c7a43bb48","side":"left"},{"sibling":"d2014279f61fb9e0da911cabc2d3a32664afdbd198aebf9c06cb2b3f9176da97","side":"right"},{"sibling":"2943aa232ba1099978a3aeee29e3e0c39faa6231597512d9c82160e7a5a2db32","side":"left"},{"sibling":"c78f0d662697b816291c956749c41d06c16e4dc4c3d390f9a90590e3f2821765","side":"right"},{"sibling":"cfb460164a914d1f96a36aa17124b45bb5417829d2adbe5ca48375dbd842ec44","side":"left"},{"sibling":"a84ebc8e894a9893ee34d1afc942d15f28243b926f1e5f9d7f10a3de1e795262","side":"left"},{"sibling":"f8f6bd9da448fa097e2115b71146f61691d9f8807aca291c87c633192a7224e9","side":"left"},{"sibling":"2dca509b3eb767a47cf215d4315f230ce9103a76264412008ae23a349b519ef1","side":"left"},{"sibling":"24d1bb4b13e0e46131b27b70a48e65fcf4e2e14b95e3bb83ade821e9df530f6b","side":"left"},{"sibling":"422bcf7e281ca3a607f3726a5e6b8fabdb85e8b32199b4a86356998e260a0b34","side":"right"},{"sibling":"54a99163a4a62374c3ca6fb46294222f1d4b1a9d0b636e27256b0e093e98239a","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":213053,"merkle_root":"19d104b92c4d7299881c447fb8422611fc9cb8615d37a538959e34a2da7ef55f","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260604T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-04T05:37:49Z","sig_algorithm":"ed25519","signature":"630748e88645187aa3b4d4cb8c872cc146180d0301e4c6656f15f99081d7776432715cea2a997b32b7b3a5216f215d19c1b536c0b093100ec843fa0bd65f2101","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_056d33977bb6065346df5464f1701ffcf3ed42ec03556ae1c85ae4ad692574c4"}}