Squaring the Circle in the Nearest Possible World: Recursive Rhombic Quadrature

The circle cannot be squared: Lindemann settled that in 1882, and nothing here disturbs his theorem. But the impossibility proof invites a sharper question: how close can a ruler-and-compass geometry legally get? We study a recursive rhombus–circle construction that generates, by itself, the family of candidate circle constants √q with q rational. Three things turn out to be provable, non-circular, and not previously assembled in one place: (i) within the integer subclass √q — the natural bounded-complexity cut, since over the dense rationals no nearest member exists — exactly one member is closest to π, namely √10, because 10 is the integer nearest π²; (ii) √10 is realized exactly as the circle constant of a genuine Minkowski norm whose unit disc interpolates the generating rhombus, with superellipse exponent p₀ = 1.7582921928…; and (iii) √10 is the unique positive constant phase-locked to the decimal system: log₁₀√10 = 1/2 exactly, so its dilation ladder never drifts against the decades, while every other constant — π included — collides with a neighbouring shell of the nested figure in finitely many steps. Along the way we do some honest bookkeeping: we exhibit the tautologies that folklore "quadratures" usually trade on, and we say precisely which steps are free choices, which are identities, and which are theorems. Includes dependency-free Python verification scripts (30 numerical checks).