Optimizing Explicit Unit-Distance Lower-Bound Certificates

Abstract: The 2026 disproof of Erdős'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 $n<sup>{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n<sup>{1.014}$ unit distances for arbitrarily large $n$ and exposes finite parameters whose choice is not fully optimized. This report formulates the finite parameter-selection task as a variant of a nonlinear integer programming problem and proposes an open-source Python verification pipeline, first validated by reproducing Sawin's published parameter choice and then applied to computationally improved certificates. The main computational contribution is an integer optimization and checking procedure for the sets of primes $T$ and $S_Q$, the integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The optimization pipelines are intentionally lightweight and replicable on standard hardware: we propose a deterministic greedy construction heuristic, a Tailored Integer Evolution Strategy with repair operators for number-theoretic feasibility, and a two-parent discrete-recombination variant. Four certificate levels are compared: Sawin's published example with $δ=0.0141144286784982\ldots$, a greedy optimization certificate with $δ=0.0151718056372133\ldots$, a Tailored Integer Evolution Strategy certificate with rational $R=6672416/100000$ and $δ=0.0152616610684193\ldots$, and a Tailored Integer Evolution Strategy with discrete recombination, again with $R=6672416/100000$, giving $δ=0.0152628688170072\ldots$. Consequently, subject to Sawin's explicit criterion being applied exactly as cited, the best current certificate supports the cautious clean statement $u(n)&gt;n<sup>{1.0152}$ for arbitrarily large $n$.

• The paper improves Sawin’s explicit unit-distance lower-bound certificate using greedy selection and a tailored integer evolution strategy, raising the verified exponent gain from 0.0141144 to 0.0152629.
• The optimization treats prime selection, multiplicities, and the parameter R as a constrained nonlinear integer program, while deterministic arithmetic checks verify primality, non-splitting conditions, budget feasibility, and high-precision values.
• The best certificate supports u(n) > n^1.0152 for arbitrarily large n, but does not construct planar point sets and leaves optimization over the quadratic-field prime set T and the gap to Sawin’s theoretical ceiling open.

The paper "Optimizing Explicit Unit-Distance Lower-Bound Certificates" (2606.03419) addresses a finite parameter-optimization problem arising from Sawin's explicit quantitative refinement of the 2026 disproof of Erdős's unit-distance conjecture. Rather than reproving the counterexample, the author treats Sawin's explicit lower-bound criterion as a nonlinear integer programming problem and applies lightweight combinatorial optimization heuristics to improve the certificate. The main outcome is a verified exponent gain $\delta = 0.0152628688170072\ldots$, improving on Sawin's published $0.0141144286784982\ldots$, which supports the clean statement $u(n) > n^{1.0152}$ for arbitrarily large $n$.

Background: the unit-distance problem and Sawin's criterion

For a finite point set $P \subset \mathbb{R}^2$, let $U(P)$ count pairs at Euclidean distance exactly 1, and define $u(n) = \max_{|P|=n} U(P)$. Erdős conjectured that lattice-type constructions are essentially optimal, i.e., $u(n)$ should be bounded by $n^{1+o(1)}$, while the best general upper bound remains $O(n^{4/3})$ via Szemerédi–Trotter-type incidence methods. The 2026 counterexample to Erdős's conjecture—announced by OpenAI and subsequently given a human-verified exposition by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, and Matchett Wood—shows that $0.0141144286784982\ldots$0 can exceed $0.0141144286784982\ldots$1 for a fixed positive $0.0141144286784982\ldots$2. Sawin's refinement made this quantitative, exhibiting more than $0.0141144286784982\ldots$3 unit distances for arbitrarily large $0.0141144286784982\ldots$4 through an explicit but not fully optimized choice of finite parameters.

Sawin's criterion reduces to selecting a set $0.0141144286784982\ldots$5 of odd primes defining a quadratic field $0.0141144286784982\ldots$6, a set $0.0141144286784982\ldots$7 of rational primes that are ramified or inert in $0.0141144286784982\ldots$8 (never split), integer multiplicities $0.0141144286784982\ldots$9 for $u(n) > n^{1.0152}$0, and a real parameter $u(n) > n^{1.0152}$1. These must satisfy a Golod–Shafarevich budget inequality

$u(n) > n^{1.0152}$2

and the objective is the exponent gain

$u(n) > n^{1.0152}$3

where the numerator collects logarithmic terms including $u(n) > n^{1.0152}$4 (with $u(n) > n^{1.0152}$5 for $u(n) > n^{1.0152}$6 or $u(n) > n^{1.0152}$7, else 1), and the denominator is $u(n) > n^{1.0152}$8. Maximizing $u(n) > n^{1.0152}$9 over admissible $n$0 is precisely the nonlinear integer program studied here.

Integer-programming formulation and heuristic methods

The paper formulates the certificate selection as a nonlinear integer optimization problem with arithmetic feasibility constraints. Two heuristic approaches are compared against Sawin's published parameters.

Greedy budget heuristic: a deterministic knapsack-style construction interprets the Golod–Shafarevich inequality as a budget, with non-splitting primes costing one unit and splitting primes costing two. Primes are ranked by benefit-to-cost ratio $n$1, multiplicities are assigned from a provisional target exponent $n$2, and a grid scan over $n$3 completes the construction. The baseline fixes $n$4 to Sawin's published prime set and considers candidate primes up to $n$5.

Tailored Integer Evolution Strategy: following Rudolph's integer mutation model based on the $n$6-symmetric double geometric distribution, this method mutates integer components directly via two-sided geometric steps rather than rounding real-valued mutations. Chromosomes encode selected prime indices, multiplicities, and the numerator of a rational representation $n$7 with fixed denominator $n$8. Problem-specific repair operators enforce distinct primes, admissible multiplicities, the permitted $n$9 range, and the Golod–Shafarevich budget before evaluation. The best variant adds two-parent discrete recombination, where each offspring component is inherited independently from one of two parents. The reported run used population size $P \subset \mathbb{R}^2$0, offspring number $P \subset \mathbb{R}^2$1, and $P \subset \mathbb{R}^2$2 generations—23,064 objective evaluations completing in under ten minutes on standard hardware—with fixed random seeds for reproducibility.

A critical design principle throughout is separation of roles: the optimization layer is heuristic, but the scoring layer is deterministic. Every reported candidate is checked independently by exact integer arithmetic covering primality, parity conditions on $P \subset \mathbb{R}^2$3, Legendre-symbol witnesses for non-splitting, budget saturation, positivity of multiplicities, and high-precision evaluation of $P \subset \mathbb{R}^2$4 using 80-digit decimal arithmetic.

Verified certificate levels

The verification pipeline was first validated by reproducing Sawin's published example exactly: with $P \subset \mathbb{R}^2$5 containing thirteen odd primes (exactly seven congruent to $P \subset \mathbb{R}^2$6), twenty-two selected primes all ramified or inert in $P \subset \mathbb{R}^2$7, and $P \subset \mathbb{R}^2$8, the pipeline reproduces $P \subset \mathbb{R}^2$9 and confirms exact saturation of the Golod–Shafarevich budget ($U(P)$0). This validation establishes confidence before applying the same checks to optimized candidates.

Certificate	Method	$U(P)$1	$U(P)$2
Sawin published	validation baseline	72	0.0141144286784982…
Greedy	deterministic improvement	66.72240803	0.0151718056372133…
Integer ES	evolutionary improvement	6672416/100000	0.0152616610684193…
Integer ES + recombination	recombination variant	6672416/100000	0.0152628688170072…

Each row passes the implemented arithmetic checks. The greedy step alone improves the displayed exponent substantially; the Tailored Integer Evolution Strategy then improves further by coordinating changes across prime selection, multiplicities, and $U(P)$3 simultaneously—changes unlikely to be found by one-pass greedy construction. Discrete recombination yields a small additional gain. The best verified certificate supports the conservative statement $U(P)$4 for arbitrarily large $U(P)$5, conditional on Sawin's criterion being applied exactly as cited.

Reproducible implementation

All code is publicly available at the author's GitHub repository, structured around three scripts: verify_all_certificates_v23.py recomputes all arithmetic checks and writes text and JSON summaries; optimize_certificates.py runs the deterministic greedy baseline; and rudolph_integer_es_discrete_recombination.py runs the improved evolutionary method. The deliberate separation between optimization and verification scripts means any reader can rerun the full pipeline—including validation against Sawin's published parameters—in minutes on commodity hardware.

Limitations and open questions

Several limitations are stated plainly by the author. First, no coordinate realization of the optimized certificate has been generated; producing actual planar point sets would require an explicit finite level of the relevant class-field tower, computable algebraic bases, embeddings, ideals, and a projection/windowing procedure—a separate computational number-theory project beyond the scope of this work. Second, the optimization fixes $U(P)$6 to Sawin's published prime set throughout; optimizing over $U(P)$7 itself remains unexplored. Third, the numerical margin over the conservative target (approximately 0.049 in absolute terms) is large relative to floating-point roundoff, but publication-grade rigor would require formal interval arithmetic and ideally independent SageMath or Magma verification. Fourth, the result depends entirely on Sawin's criterion being applied exactly as cited; the author explicitly notes they are not a specialist in algebraic number theory and use Sawin's preprint as the mathematical basis without independent proof of the underlying theorem.

Most significantly, the achieved $U(P)$8 remains far below the abstract upper bound of $U(P)$9 for the exponent obtainable from Proposition 15 of Sawin's formulation. The computational optimization space within this framework is therefore far from exhausted, and it is an open question how much of the remaining gap can be closed by better parameter search versus stronger number-theoretic estimates.

Conclusion

This paper demonstrates that modern but deliberately lightweight integer optimization heuristics—a deterministic greedy construction and a Tailored Integer Evolution Strategy with repair operators and discrete recombination—can improve explicit certificates in pure mathematics when the certificate problem admits a finite integer-programming formulation. The best verified certificate achieves $u(n) = \max_{|P|=n} U(P)$0, supporting $u(n) = \max_{|P|=n} U(P)$1 for arbitrarily large $u(n) = \max_{|P|=n} U(P)$2 at the level of verified finite certificate selection, an improvement over Sawin's displayed $u(n) = \max_{|P|=n} U(P)$3. The contribution is computational and methodological rather than a new coordinate construction, and the openly available implementation invites further optimization toward the theoretical ceiling suggested by Sawin's abstraction.