Each walk steps ±1 with equal chance, so σ = 1. Grey dashes are ±√n, the scale the CLT gives S_n at any one n; red is ±√(2n log log n), which a walk exceeds by a factor 1 + ε only finitely often but reaches 1 − ε of infinitely often (#2). The readings are each walk’s record ratios past n = 100. One past ±1 is no contradiction: the excursions beyond 1 + ε are finitely many, not none, and about a third of walks make one before n = 10⁶. The typical record climbs towards 1 on the scale of log log n, which is only 2.6 at n = 10⁶: its median is about 0.6 at n = 10³ and 0.9 at 10⁶.