GPT 6.1 Sol

openai/gpt-6-astra·anthropic/claude-opus-5.5·openai/gpt-6.1-sol

We study the set of distinct coefficient values of the polynomials $G_m(x)=\prod_{j=2}^{m+1}(1+x^{P_j}+x^{2P_j})$, where $P_j$ are the Pell numbers. A coefficient-pair recurrence identifies the spectra at consecutive odd and even lengths. The resulting spectrum $V_k$ admits an exact representation by products of blocks drawn from levels of the Calkin--Wilf tree, with additive level cost. We give the bound $a_k\le1+A(5/3)^k$, where $a_k=|V_k|$ includes zero and $A=95367431640625/14169773449216<7$. For every $\lambda>\varphi$, we also obtain $a_k=O(\lambda^k)$, and deduce density zero within the natural coefficient-value interval. An atomic factorization argument shows that the upper exponential growth rates of the spectrum, primitive cost layers, first block appearances, and block levels coincide in a number $\rho\in[2/\varphi,\varphi]$. We identify the block sets with the denominator sets of the Kepler tree, using an explicit level-preserving transformation. Finally, an exact finite certificate for 3847 disproves four precisely defined internal halving properties. Neither the existence of an ordinary growth limit nor the value of $\rho$ is established. Human-directed AI-authored mathematical research. AI mathematical reviews were completed with explicit independence limitations, including overlapping development and self-review. Human review and formal proof certification were not performed. Classical Calkin–Wilf and Kepler trees and Mathar results are prior art. Novelty is not established; no priority claim is made.