Marcin

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-luna·anthropic/claude-opus-5.5

Let $P_n$ be the Pell numbers and, for $\ell\ge1$, let $\rho_\ell(n)$ be the coefficient of $x^n$ in $G_\ell(x)=\prod_{j=2}^{\ell+1}(1+x^{P_j}+x^{2P_j})$, that is, the number of words $(d_2,\dots,d_{\ell+1})\in\{0,1,2\}^\ell$ with $\sum_j d_jP_j=n$. We prove that $B_\ell=\max_n\rho_\ell(n)=2^{\lfloor(\ell-1)/2\rfloor}$ for every $\ell\ge1$, and we determine the set $M_\ell$ of all maximizers for every length. For odd $\ell\ge3$ it consists of the four integers $u_\ell,\ u_\ell+2,\ u_\ell+P_{\ell+1},\ u_\ell+P_{\ell+1}+2$ with $u_\ell=(P_{\ell+2}-5)/2$, and $M_1=\{0,2,4\}$. For even lengths $|M_2|=9$ and $|M_{2k}|=16k-8$ for $k\ge2$; moreover $M_{2k}$ is the union of two explicit six-element cores and four explicit chains of two-element blocks. All proofs rest on one structural fact: when two consecutive Pell weights are adjoined, the nine shifted copies of the previous coefficient sequence overlap at most two at a time. As an application, let $\delta=1+\sqrt2$ and let $\nu_m(t)$ be the number of strings $(d_k)_{k=-2m}^{2m}\in\{0,1,2\}^{4m+1}$ with $\sum_kd_k\delta^k=t$. We prove the exact identity $\nu_m(t)=\rho_{4m+1}(tP_{2m+2})$ for every integer $t$. Consequently $\max_{t\in\mathbb Z}\nu_m(t)=4^m$, attained only at $t=\sum_{j=-m}^{m}\delta^{2j}=P_{2m+1}+P_{2m}$ when $m\ge1$ and at $t\in\{0,1,2\}$ when $m=0$, and every fibre over $\mathbb Z[\sqrt2]$ in the same window contains at most $4^m$ strings. The results and proofs constitute human-directed, AI-authored mathematical research; they have been checked only by AI-model audits; no independent human proof review has taken place, and novelty has not been established.

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.