The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum
Let $φ=(1+\sqrt5)/2$. The set $\{0,φ,φ^2,φ^3\}$ is classical (Euclid, Elements XIII.8): by $φ^2=φ+1$ its six pairwise differences take the values $1,φ,φ^2,φ^3$, with multiplicities $1,2,2,1$. We write it as the golden quadruple $S=(F, F-a, F-b, F-a-b)$, $a=φu$, $b=φ^2u$, $F>a+b$, and study its extremal and rigidity properties. No five distinct reals have all their positive differences in $\{uφ^n:n\in\mathbb{Z}\}$; this follows from a result of Lenstra (1977), and we give a self-contained proof. We give an elementary proof of uniqueness: every four-element set with this property is a translate of $uφ^m\{0,φ,φ^2,φ^3\}$ for some $m\in\mathbb{Z}$, and if all its differences lie in $\{u,φu,φ^2u,φ^3u\}$ it is a translate of $S$. Among all $x>1$, only $x=φ$ admits a four-point set of reals whose set of distinct differences is $\{c,cx,cx^2,cx^3\}$. For the analogous quadruple with base $x\neqφ$ the failure is measured by two closure residues, which along Fibonacci quotients equal $\pm u/\mathrm{Fib}_n^2$ and $\pm \mathrm{Fib}_{n+1}u/\mathrm{Fib}_n^3$. The cosine sum $C_S(t)=\sum\cos(2πd t)$ over the six differences equals $8\cos^2(πa t)\cos^2(πb t)-2$; it is not periodic but is Bohr almost periodic, with explicit almost periods $τ_n=\mathrm{Fib}_n/u$, and we determine the two-term asymptotics $\sup_t|C_S(t+τ_n)-C_S(t)|=6\sqrt3 πφ^{-n}-2\sqrt3 π^3φ^{-3n}+O(φ^{-5n})$, with optimal constants; the inclusion length of the $\varepsilon$-almost periods is of order $1/\varepsilon$. For a scale $K>0$ the golden ladder $ν_n=Kφ^{-n}$ realises the construction with differences $ν_{200},\dots,ν_{203}$; we evaluate it for $K=\sqrt{c^5/(Gh)}$. All proofs are elementary and self-contained.