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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 treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $\\delta=0.014114\\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $\\delta=0.015263\\ldots$ and supporting th","title":"Optimizing Explicit Unit-Distance Lower-Bound Certificates","url":"https://arxiv.org/abs/2606.03419","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.03419v5 Announce Type: replace-cross \nAbstract: The 2026 disproof of Erd\\H{o}s's unit-distance conjecture and Sawin's 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 treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $\\delta=0.014114\\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $\\delta=0.015263\\ldots$ and supporting th","title":"Optimizing Explicit Unit-Distance Lower-Bound Certificates","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-10T04:43:37Z","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:a1dfa92daef3ef763527caf6ab6163eff53d818db90c7c5e067e4dc7bac0a7b0e43046acf8cb62374c632ae598e44da48d5aba9507d96e65ad4c909a9afaa608","signer":"crovia.substrate","subject":{"observed_at":"2026-06-10T04:43:37Z","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":"dada7b084f6f161fc80119293e0e1e40726dc4e04d2617cf23c568c33f2aaf38","leaf_index":226227,"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":"0ae4f7b3c380d56a78574811a59e2cf7caf48ac100d0c507660315417c5f200b","side":"left"},{"sibling":"5298b313db488e6190ad60606060420692b068dbc8c74fb4f3f91b1dea083688","side":"left"},{"sibling":"cd411bac6b174b2d6a4f9637af6567e98b3c72646a49baf714ec9925b64c688e","side":"right"},{"sibling":"33b2492b6db752dacfd47e08fe0708cd0a15055f76161f0150fe49c81a663425","side":"right"},{"sibling":"e53d5862d7e24220e12cf9b6e10480bdcebd83380bd70a1a12cd2734d10afb42","side":"left"},{"sibling":"2093011af3a761103a212129070a946c21ea019691ee2b38d5b3785613d4f937","side":"left"},{"sibling":"a026ba4444dd60937f3a5636d15324f239921221e1299500edf326142aa1039a","side":"right"},{"sibling":"891febe0a62df3512be2c4743bc3c5d3327de005faf591d59848778f0454f464","side":"left"},{"sibling":"b180cfc3f8638c912e90139ec42e2e3fd9b67b3fe342b36e8954314633ae5f61","side":"left"},{"sibling":"a4d17aefe58175050dc159af6246658fcf1c9f3ed57aacf1b350fc3261de4e69","side":"left"},{"sibling":"280b980aa0c7756b0b0cb22658f26466d36f0e70fbc3312cd2311d9898e30b8f","side":"right"},{"sibling":"c98954d4b658b1dda60fe52576fcf9bf21a2d49c67fb63f8c30f16ab5f721938","side":"right"},{"sibling":"cdb58f86163046d3b15f857b03372ec75e1ad9ea4548e086793d528b9eed364d","side":"left"},{"sibling":"533d82482604463aa4a281b9d8b85917b383b7c5f924b2b494039524c55e8797","side":"left"},{"sibling":"b2590791b920ca2a4ed39de126d2c0b1a10d9e7e62f572f12425f214e767b6e1","side":"left"},{"sibling":"6cea4964f32722eb370847c2f7c9d6a9f0622c239538b07e6815a59d6fd8d49c","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":228173,"merkle_root":"7e416202c0bfd759bd2eea4236713b403993d99793fe8badb5065040080bece3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260611T143708Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-11T21:59:35Z","sig_algorithm":"ed25519","signature":"231c80024bc3982dd493c45b31af95097e97aabc6d712a4e5bad7d0cbdd3c08e01ff395b0f8e72754bac97016e0cd0eed88b8a13cb71edbbcb9b6d72c10a7b03","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_341ff53699a92f302417e2a6f2084739b6637d0fcf35ae945e478ae6ec064456"}}