Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.
Date: 23 September 2026. Source versions read in full: 757.tex in gpt_pro_5.4.
1) OBJECTIVE AND FINITE CHECK
A weak $(4,5)$ set of real numbers has at least five distinct positive distances on every four-element subset. The source reduces its strong $(3,3)$ subsets to sets containing no three-term arithmetic progression, and supplies \[
A=\{0,136,200,243,246,249,272,286,298,323,400,528,596,1056\}.
\] I checked all $\binom{14}{4}=1001$ four-element subsets using integer arithmetic. Every one has at least five distances. There are exactly twelve arithmetic-progression triples. Enumerating all $2^{14}$ subsets gives maximum progression-free size eight. These checks are reproduced in verification/root_checks.py.
2) REPLICATION WITHOUT CREATING EXTRA RELATIONS
Ordinary translated copies fail: equal internal distances in two copies violate the four-point requirement. Instead fix a positive transcendental real $t$ and form \[
B_k=\bigcup_{i=1}^k\{t^{2i}+a t^{2i+1}:a\in A\}.
\] Different listed points are distinct, because a nonzero integer polynomial cannot vanish at $t$. Any equality of positive distances can be written $x-y=\pm(z-w)$. For four distinct endpoints in more than one block, comparison of the coefficients of each $t^{2i}$ forces each occupied block to occur equally often on the two sides. The only mixed possibility is two blocks each containing one point on each side; comparison of $t^{2i+1}$ then forces those points to coincide, a contradiction. For three distinct endpoints a repeated distance means $2x=y+z$. Its constant-block coefficients force all three into the same block. Therefore all nontrivial distance repetitions are internal. Four points in one block inherit the property of $A$; three in one block and one outside have at most the one distance repetition possible on three real points; every other mixed four-set has all six distances distinct. Thus $B_k$ is weak $(4,5)$.
3) EXACT INDEPENDENCE RATIO
The same coefficient argument shows that every arithmetic progression in $B_k$ lies in a single block. A progression-free subset has at most eight points in each block, and independent choices of eight-point progression-free subsets attain this bound. Consequently \[
|B_k|=14k,\qquad \alpha_{\rm AP}(B_k)=8k.
\] The finite $4/7$ obstruction therefore persists at arbitrarily large sizes, rather than being an isolated fourteen-point effect.
7) FINAL
UNRESOLVED. This verifies and amplifies the supplied example. It does not close the source's gap between the general lower bound and $4/7$, nor claim a new best upper bound.
8) COMPLETION ESTIMATE
COMPLETION: 15% This is a subjective estimate of progress toward the entire stated problem, not a probability that the conjecture or an unverified proof is correct.