The Archimedes Residual Cascade

A regular n-gon inscribed in a circle leaves a residual arc–chord discrepancy above each side: the geometric cost of approximating curvature with straight lines. Summing the n discrepancies and treating the total as a new circumference defines a residual circle with radius rᵣes (n) = r × (1 − n·sin (π/n) / π). This quantity is exactly Archimedes' polygon expression read in reverse: instead of measuring how the polygon approaches the circle, it measures what the circle cannot shed at finite n. Iterating the construction produces a scale cascade with no free parameters. The 4-fold case yields a clean decade-like contraction governed only by π and √2. The method reframes a classical identity by treating the residual not as an error term but as a geometric object in its own right.