GPT 6 Astra

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.

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

Let $\varphi=(1+\sqrt5)/2$ and, for $m\ge 0$ and an integer $t$, let $\mu_m(t)$ be the number of subsets $S\subseteq\{-2m,\dots,2m\}$ with $\sum_{k\in S}\varphi^k=t$. We prove that $\mu_m(t)=R(F_{2m+2}\,t)$ for $0\le t\le L_{2m+1}$, where $R(n)$ is the number of representations of $n$ as a sum of distinct Fibonacci numbers, and that $\mu_m(t)=0$ for all other integers $t$; the correspondence is an explicit shift of exponents, and the proof uses only the algebraic conjugation of $\mathbb{Z}[\varphi]$. Combining this transfer with a symmetry and a forced-digit argument, we reduce the extremal problem for $\mu_m$ to results of Koc\'abov\'a, Mas\'akov\'a and Pelantov\'a on the maxima of $R$ over intervals of fixed Zeckendorf length. We obtain $\max_{t\in\mathbb{Z}}\mu_m(t)=F_{2m+1}$ for every $m\ge 0$, and we show that for $m\ge 2$ the maximum is attained exactly at $t\in\{F_{2m},F_{2m+1},2F_{2m},F_{2m+2}\}$. We compare the transfer with a theorem of Dekking and van Loon on natural base-$\varphi$ expansions. The results and proofs constitute human-directed, AI-authored mathematical research. They have been checked only by AI-model audits; no independent human mathematical verification has been completed, and novelty has not been established.