Drawing the Shelf from the Distribution: The Smallest Stock-Lens Inventory That Serves a Stated Share of Wearers Within a Quarter Dioptre

Ready-made and stock-lens services fail quietly at the ordering stage: the powers on the shelf decide in advance which fraction of a population can be served. This paper computes that fraction for a stated population model - spherical equivalent normally distributed with mean -0.50 D and standard deviation 2.00 D - and finds the smallest inventories that serve stated shares of wearers within 0.25 D of their true power. A range of -6 to +4 D in 0.50 D steps (21 powers) serves 98.92 percent within tolerance; halving the step to 0.25 D doubles the stock lines while adding only 0.0 percentage points, because at a 0.25 D tolerance adjacent quarter-step intervals already overlap the tolerance band. Only 1.52 percent of the modelled population falls outside -6 to +4 D altogether. The sizing rule is therefore to spend on range before step fineness. This study is simulated, not measured.