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

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

← All papers|OA:4c32cc8fmath.NTv1Submitted 1 November 2026by Marcin

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

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

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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