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+√5)/2. The four-point set {0, φ, φ², φ³} is classical: up to a dilation it consists of the two endpoints and the two crossing points on a side of the regular pentagram (Euclid, Elements XIII.8), and, by the identity φ² = φ + 1, its six pairwise differences are 1, φ, φ², φ³, with multiplicities 1, 2, 2, 1. We write this configuration as the golden quadruple S = (F, F−a, F−b, F−a−b), where u > 0, a = φu, b = φ²u and F > a+b, and study its extremal and rigidity properties. We prove that it is maximal and unique: no five distinct real numbers have all their positive differences in {u, φu, φ²u, φ³u} (indeed, not even in the whole set {uφⁿ : n ∈ ℤ}), and every four-element set with this property is a translate of S. We prove that the golden ratio is forced: φ is the only ratio x > 1 for which some four-point set of reals has the four-term progression {c, cx, cx², cx³} as its set of distinct differences. For the analogous quadruple with base x > 1, x ≠ φ, the failure is measured by two closure residues, (x²−x−1)u and x(x²−x−1)u, which along Fibonacci quotients equal ±u/Fₙ² and ±Fₙ₊₁u/Fₙ³. Finally, the cosine sum C_S(t) = Σ cos(2πdt) over the six differences equals 8cos²(πat)cos²(πbt) − 2; it is not periodic, but it is almost periodic in the sense of Bohr, with explicit almost periods Fₙ/u. For an arbitrary scale K > 0 the golden ladder νₙ = Kφ⁻ⁿ realises the construction with u = ν₂₀₃, so that the differences are ν₂₀₀, …, ν₂₀₃; we evaluate it for K = √(c⁵/(Gh)). All proofs are elementary and self-contained, and the main theorems are formally verified in Lean 4 with Mathlib. We state explicitly what is classical and what is new, in relation to difference sets, Golomb rulers, Sidon sets, Fibonacci numbers, quasicrystals and almost periodic functions.