Emmerich 2026 optimizing explicit unit distance lower bound certificates

../

proposition_1: Records that Emmerich's verification pipeline, run on Sawin's published data with R = 72, reproduces Sawin's exponent delta = 0.0141144287 to the digits shown.

proposition_2: Records Emmerich's re-optimized certificate data with Sawin's prime set T and the conditional bound u(n) > n^1.0152 for arbitrarily large n.

---

Michael T. M. Emmerich, Optimizing explicit unit-distance lower-bound certificates. arXiv:2606.03419v5 [math.OC], 9 June 2026, 20 pages.

Version read and provenance

The copy read for this card is the arXiv v5 manuscript; its p. 1 watermark reads "arXiv:2606.03419v5 [math.OC] 9 Jun 2026". Provenance: 513,718 bytes, downloaded from https://arxiv.org/pdf/2606.03419v5 on 2026-09-05. The arXiv record is https://arxiv.org/abs/2606.03419. The text refers to its own versions 3 and 4 (pp. 2 and 9); no earlier version, later version, or publication was acquired or checked at filing, which had no network access. Result citations name v5 pages. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2606.03419), every other right reserved.

What the report does

The report treats the finite parameter choice in Sawin's explicit criterion as a nonlinear integer optimization problem over data $(T,S_{\mathbb Q},k,R)$ and supplies a Python optimization and verification pipeline (pp. 1--2, 4). Its exponent formula (1) on p. 4,

$$\delta(T,S_{\mathbb Q},k,R)=\frac{N(T,S_{\mathbb Q},k,R)}{D(T,S_{\mathbb Q},k,R)},$$

with $e(p)=2$ for $p=2$ or $p\in T$ and $e(p)=1$ otherwise, is the exponent gain recorded as equations (12)--(13) on the [[discrete_geometry/sawin_2026_explicit_lower_bound_unit_distance_problem/theorem_1|Sawin Theorem 1 page]]. The report states on p. 2 that its results "are based solely on Sawin's 2026 paper and on the optimization problem explicitly stated there, together with direct extensions of the prime range $T$" and that the MathOverflow and Zenodo certificates it mentions (values above $\delta>0.035$ and $1+\delta=1.03158935$, pp. 1--2) are cited as related work, not incorporated into its verified certificates.

Its three results, all with Sawin's set $T=\{3,5,7,\ldots,43\}$ unchanged:

• [[discrete_geometry/emmerich_2026_optimizing_explicit_unit_distance_lower_bound_certificates/proposition_1|Proposition 1]] (p. 14, "Validation against Sawin's published example"): the pipeline reproduces Sawin's published data ($S_{\mathbb Q}$ of 22 primes, the multiplicities $k(p)$, $R=72$) and the value $\delta=0.014114428678498239\ldots$ (numerator $3.8822487482003876\ldots$, denominator $275.0553236430010\ldots$). The denominator and $\delta$ agree with the rational enclosure (15) on the Sawin Theorem 1 page after rounding to the digits shown; the numerator agrees with (15), $3.88224874820038782\ldots$, to fifteen decimals, but its sixteenth decimal is printed as $6$ where (15) has $8$.
• The re-optimized certificates (Table 1, p. 8; data on pp. 14, 17--18): a greedy certificate with $\delta=0.0151718056\ldots$, a Tailored Integer Evolution Strategy certificate with $\delta=0.0152616610\ldots$, and a discrete-recombination variant with $\delta=0.0152628688\ldots$, the abstract's "0.015263...". The last two share $R=6672416/100000$ and the 22-prime set $S_{\mathbb Q}$ recorded on [[discrete_geometry/emmerich_2026_optimizing_explicit_unit_distance_lower_bound_certificates/proposition_2|Proposition 2]], and differ only in three multiplicities.
• [[discrete_geometry/emmerich_2026_optimizing_explicit_unit_distance_lower_bound_certificates/proposition_2|Proposition 2]] (p. 17, "Lower-bound consequence"): assuming Sawin's explicit criterion is applied exactly as in Sawin's paper, the Tailored Integer Evolution Strategy certificate supports $u(n)>n^{1.0152}$ for arbitrarily large $n$, where $u(n)$ is the maximum number of unordered unit pairs among $n$ planar points (p. 2).

The abstract and Section 8 (pp. 1, 10) also report an Emmerich--Cordella certificate for an extended prime range, $\#T=67$, with $u(n)>n^{1.031}$, deposited on Zenodo on 6 June 2026 (reference [10], p. 20). That certificate's data are not printed in this report and its Zenodo record is not held.

Source scope and limits

The report's own limits are recorded with the result. It says "No claim is made here that a coordinate realization of the optimized candidate has been generated" (p. 3); its verifier evaluates the transcendental terms in "high-precision decimal arithmetic" (p. 13; 80 digits, p. 17) and says "a fully formal proof certificate would ideally replace the final floating-point step by interval or rationally certified bounds" (p. 13); and Remark 2 (p. 19) says the sharper decimals "should be treated as candidate decimals until independently checked with interval arithmetic and reviewed by a human expert in the number-theoretic construction". The printed admissibility-witness table (p. 16) is for the greedy certificate's prime set; for the two evolution-strategy certificates the report says the same checks pass (pp. 17--18) without printing witnesses. The Declaration on p. 20 says that "OpenAI ChatGPT 5.5 was used as an auxiliary tool for programming assistance, code review, debugging, and cross-checking the interpretation and implementation of the constraints and optimization model" and that AI tools "were not used to design the optimization algorithms, generate mathematical proofs, generate the certificates, prepare the related work discussion, or select references".

Read status: Proposition 1, Table 1, Proposition 2 and the two certificate displays on pp. 17--18 were read clause by clause on the page images (claims checked); pp. 1--2, 8--10 and 19--20 were read; the algorithmic material of Sections 4 and 6--7 and Appendix A was inspected for structure only. The report's certificates were not replayed, its code was not run, and no independent check of the new $\delta$ values exists here; the exponent $1.0152$ stands as the source's conditional claim. Nothing here is a Lean proof or formal verification.

Bears on. Problem 90: Proposition 1 reproduces the exponent $\delta\approx0.0141144$ of Sawin's Theorem 1 from Sawin's data; Proposition 2 reports, with Sawin's prime set $T$, a certificate supporting $u(n)>n^{1.0152}$ for arbitrarily large $n$, assuming Sawin's criterion is applied exactly as in Sawin's paper, and Remark 2 (p. 19) calls its decimals candidates until independently checked. The disproof does not depend on either, since any fixed positive exponent gain already exceeds every $C/\log\log n$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.