The Axis Budget: How Many Degrees of Cylinder Misalignment Each Cylinder Power Can Afford

When a cylindrical lens is rotated away from its prescribed axis, it does not simply weaken: it leaves a new, oblique astigmatism whose size is twice the cylinder power times the sine of the axis error. This paper turns that standard vector result into a fitting tolerance table for workshops without lensmeter verification at every step. Holding residual astigmatism to a stated 0.25 D requires the axis of a 1.00 D cylinder to be set within 7.2 degrees, a 2.00 D cylinder within 3.6 degrees, and a 3.00 D cylinder within 2.4 degrees. A common shop habit of accepting anything inside ten degrees is shown to be harmless for small cylinders and clearly inadequate above about 1.50 D. A low-cost safeguard follows: verify axis against the prescription for every cylinder of 1.50 D or more before the pair leaves the bench. This study is simulated, not measured.