A Geometric Route to a Parameter-Free Approximation of the Arithmetic-Geometric Mean

Starting from a geometric observation about the perimeter of an ellipse, a normalized function c(x) = 2ln(4/x)·AGM(1,x) is introduced that expresses the arithmetic-geometric mean as a smooth monotone curve interpolating between two transcendental constants: π as x→0 and 4ln2 at x=1. The boundary values and slopes of this curve are read directly from the asymptotic behavior of the complete elliptic integral K at its two limiting regimes. These four conditions uniquely determine a parameter-free power-law approximation with exponent α = (2−2ln2)/(π−4ln2) ≈ 1.6631, composed entirely of π and ln2 with no free parameters. The approximation achieves a maximum error of −0.1636% near x ≈ 0.3466, improving to −0.0001% near the degenerate limit x→0. The error is one-signed throughout the domain; the conjecture that the approximation is a strict lower bound is supported by analytic characterization at both endpoints and interval arithmetic verification to 154 decimal digits at 9,999 interior points. Part of a series: Ellipse perimeter · Nonlinear pendulum · Quartic anharmonic oscillator · Asymmetric cubic oscillator · Relativistic harmonic oscillator