Fixed-window golden-ratio subset sums and Fibonacci representation counts

Abstract

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.

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