GPT 6 Astra Ultra attempt on Math-692

Problem 148: decoding the splitting tree and retaining the lower-order gain

1) OBJECTIVE AND SOURCE REVIEW

Attempt dated 2026-09-23. The target is the growth of $F(k)$, the number of representations of 1 by $k$ distinct positive unit fractions. I read gpt_pro_5.4/148.tex fully. Its two-phase splitting construction is useful, but the assertion that two different persistent denominators force different final sets needs their positions in the ordered list to be specified. I supply that decoding and optimize the quantitative bound without discarding two powers of 2 at every branching step.

2) WORK

Start with $(2,3,6)$ and replace a largest denominator $L$, using $d\mid L$ and $d<L$, by \[ L+d,\qquad L'=L(L+d)/d. \] Both exceed $L$ and $L+d<L'$. Thus every frozen denominator is smaller than every denominator created later. In the final increasing list the frozen denominators occur in chronological order. Given the fixed initial list and a final list, the first newly frozen entry recovers $d=(L+d)-L$; it also recovers $L'$. Induction recovers the entire sequence of choices. This is a full injectivity proof, not just persistence of individual entries.

After $r$ forced choices $d=1$, the largest denominator $M_r$ satisfies $M_0=6$, $M_{j+1}=M_j(M_j+1)$. Coprimality of the factors gives $\tau(M_r)\ge2^{r+2}$. All proper divisors of $M_r$ remain valid in each later step, since the current largest denominator remains a multiple of $M_r$. The decoded branching tree therefore proves \[ F(3+r+s)\ge(2^{r+2}-1)^s. \] Put $u=\lfloor(k-1)/2\rfloor$, $r=u-2$, and $s=k-1-u$. For $k\ge7$ these are nonnegative, $u\ge3$, and $s\le u+1$. Consequently \[ (1-2^{-u})^s\ge1-s2^{-u}\ge\frac12 \] by Bernoulli's inequality. Since $us=\lfloor(k-1)^2/4\rfloor$, we obtain the explicit refinement \[ F(k)\ge2^{\lfloor(k-1)^2/4\rfloor-1}\qquad(k\ge7). \] The leading quadratic exponent agrees with the supplied construction; the gain is a rigorously retained linear-order term, not a new growth scale.

3) ADVERSARIAL VERIFICATION

Different branches might contain some of the same numbers later, so persistence alone is insufficient; their ordered ranks are what ensure injectivity. All fractions remain distinct because both children exceed the replaced maximum. The estimate concerns a subset of all representations and gives no matching upper bound.

4) FINAL

UNRESOLVED. The splitting family has been rigorously decoded and its bound modestly sharpened. Determining the asymptotic growth of all representations remains open in this attempt; no improvement over the strongest literature bounds is claimed.

5) SOURCES CHECKED

Complete supplied file, read 2026-09-23. The splitting, divisor count, and optimization are proved directly without importing an external asymptotic estimate.

6) COMPLETION ESTIMATE

Subjective progress toward the estimation problem. COMPLETION: 12%