An optimized explicit exponent for the unit-distance problem within Sawin's construction

Sawin (arXiv: 2605. 20579) recently made explicit the OpenAI disproof of the Erdős unit-distance conjecture, proving that for arbitrarily large n there exist n-point planar sets with more than n¹. 014 unit distances, via Golod–Shafarevich class field towers over CM number fields of large degree and small root discriminant. Sawin states that his parameters were not carefully optimized and lists a search over (T, SQ, k, R) as a route to improvement. Carrying out exactly this optimization within the multiquadratic specialization underlying Sawin's Theorem 1 (his Eq. (11) and Lemma 12), and changing no mathematics, we certify an explicit exponent u (n) > n¹. 031. At Sawin's own 13-prime ramification set, a re-selection of the packet set SQ already raises the exponent to n¹. 015268 — the same level obtained independently and concurrently by Emmerich (arXiv: 2606. 03419). Optimizing the number #T of ramified primes as well (over the no-split, discriminant-minimal parity-corrected first-primes family), the exponent does not peak at #T=13 but keeps climbing to a certified theorem point at #T=50 (R = 3319777/100000, 549 packet primes, Golod–Shafarevich tight: 50+549+0+1 = 600 = floor (49²/4) ), with 1 + δ = 1. 0310580345378225582727517…; a complete scan over 13 ≤ #T ≤ 56 with checkpoints to #T=120 locates the best value found in this scan at #T=56 (n¹. 031342), after which all displayed checkpoint values remain below it. All elementary hypotheses are verified five independent ways (standard-library Python, SymPy/Euler/Pari, Pari/GP at 40 digits, and Magma at 30 digits), with the full 549-prime certificate and splitting/inertia witnesses tabulated in an accompanying supplement and the scan plotted in Supplementary Figure S1. The gap to Sawin's unconditional ceiling n¹. 24295 (his Proposition 15) is shown to require new mathematics — lower bounds for relative class numbers, with the Siegel-zero obstruction of his Remark 14 — rather than further parameter search.