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Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report starts from Sawin's nonlinear integer optimization problem and develops an open-source Python verification pipeline. The pipeline is first validated by reproducing Sawin's published parameter choice and is then applied to computationally improved certificates. We optimize and verify certificates involving sets of primes $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The implementation is deliberately lean, so that all results can be replicated on standard hardwa","title":"Optimizing Explicit Unit-Distance Lower-Bound Certificates","url":"https://arxiv.org/abs/2606.03419","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.03419v2 Announce Type: replace-cross \nAbstract: The 2026 disproof of Erd\\H{o}s's unit-distance conjecture and Sawin's subsequent explicit quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\\varepsilon}$ for a fixed positive $\\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report starts from Sawin's nonlinear integer optimization problem and develops an open-source Python verification pipeline. The pipeline is first validated by reproducing Sawin's published parameter choice and is then applied to computationally improved certificates. We optimize and verify certificates involving sets of primes $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The implementation is deliberately lean, so that all results can be replicated on standard hardwa","title":"Optimizing Explicit Unit-Distance Lower-Bound Certificates","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/2606.03419"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:4122a4512df764910e9d9a0f3333ec164fddc620f185051af249fd5517e4b28ce70052ba741059358f2ed1a9bb6e793dff07e52838b8723cd6732611a5188a0b","signer":"crovia.substrate","subject":{"observed_at":"2026-06-04T04:43:08Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.03419"},"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":"8f91dbde4f7f4d8e14340dc9e6644936ba1c24080570a57c5bf151df6bd9e871","leaf_index":212925,"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":"482de1b61170d942066dfa18d96b3ab733a580ba4d3a50ce97b3653cb6f95422","side":"left"},{"sibling":"f951e8fa53f0ff26cd4bfd550163e5e7ac30f2c65c319be8c46792a3d9d35690","side":"right"},{"sibling":"ba4104e69fd137a57eb46c14578e3e3e131c4628742cecd098e1851cb96eb4cc","side":"left"},{"sibling":"74f4281f8ca481b165f5f32194c49440df3fe2b3415239ad71d11c00d68cc04c","side":"left"},{"sibling":"c74691f599c3abba5ee129d751ba42d12be527d4dc3ba6a14a8d1ea103453091","side":"left"},{"sibling":"5c7350e870578d965e575bef6a69c5228063f3671d81819847c9c03c778993a9","side":"left"},{"sibling":"d3e42a6c46328ee67e86cd8ee47af0826f2c502d9570e347a543ab25d8fd2fee","side":"right"},{"sibling":"8e52d8857251092c8d86330e4f43c6472257636abff3006310489f2170e2b3e3","side":"left"},{"sibling":"b4e8a5af3a9fdc4bb815270cd541ee4baf09fb1d9e2cccd75c7346558e1f1e4c","side":"left"},{"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_7d0885c67c140a7e2bd0d28778a60e00f5a77bb41d53484d5071e1755ee15918"}}