The three classical Greek construction problems — squaring the circle, doubling the cube, and trisecting the angle — were proved impossible in the nineteenth century. The impossibility of each was established. The magnitude of each was not. This paper derives a precise geometric residual for each problem: the gap between the ideal ratio sought and the best constructible approximation. The three residuals are not independent — they form a structured hierarchy governed by a single base quantity, ΔG = φ − (4π/3) ^ (1/3) = 0. 006, the volumetric irresolution between the golden ratio and π in three dimensions. A secondary result distinguishes, via the origami axioms of Huzita and Hatori, which impossibilities are tool-limited and which are structurally permanent. One observation is noted without physical interpretation: ΔG equals, to seven significant figures, the correction separating the golden angle from the inverse fine structure constant. Independent researcher. Not peer reviewed.
Gregory Robert Schlimmer (Sat,) studied this question.