GPT 6 Astra Ultra attempt on Math-755
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Erdős Problem #312: independent continuation
Date: 23 September 2026. Source versions read in full:312.tex in gpt_pro_5.4.1) OBJECTIVE AND SOURCE AUDIT
The target is exponential closeness to one from below when a multiset has reciprocal mass greater than $K$. The source's maximal-subset argument gives only inverse-linear closeness. I improve its elementary counting constant using dyadic chains; this still does not match even the known inverse-square bound, let alone the conjectured exponential rate.2) WORK: A CHAIN HAS TOTAL MASS LESS THAN ONE
If a multiset contains denominators only of the form $m2^j$ for a fixed odd $m$, and has total reciprocal mass at least one, it has a submultiset of mass exactly one. To see this, choose the largest exponent $J$ present and scale by $m2^J$. The weights are powers of two at most $2^J$, and the target is the multiple $m2^J$. Repeatedly combine pairs of equal smaller weights until every smaller denomination has at most one copy. Those residual weights total less than $2^J$. If total mass is at least $m2^J$, at least $m$ bundles of weight $2^J$ must have been formed. Taking $m$ of those disjoint bundles proves the claim.Now choose a submultiset $S$ with maximal mass $1-\delta\le1$. If $\delta>0$, there is no exact-one submultiset anywhere. Every omitted denominator is less than $1/\delta$, by maximality. Put $M=\lceil1/\delta\rceil-1$. Partition the omitted denominators according to their odd parts. There are at most $\lceil M/2\rceil$ chains, and each has mass strictly less than one by the preceding lemma. Therefore \[ K-1+\delta<R(A\setminus S)<\lceil M/2\rceil \le(M+1)/2<(1/\delta+1)/2. \] After rearrangement this gives \[ 2\delta^2+(2K-3)\delta<1, \qquad \delta<\frac{\sqrt{(2K-3)^2+8}-(2K-3)}4. \] For large $K$ the right side is $(1+o(1))/(2K)$, improving the source's elementary leading constant by a factor of two.