We analyze the iteration of the harmonic mean with one fixed endpoint and show that it admits an exact reciprocal linearization. The nonlinear recurrence H_{n+1} = \frac{2aH_n}{a+H_n} is transformed by the substitution u_n = 1/H_n into a linear affine map with universal contraction rate 1/2, independent of the initial values. This yields a closed-form expression for the full sequence and establishes exponential convergence to the fixed point a. We show that the iterated sequence contains strictly more information than the single harmonic mean: while H(a,b) constrains (a,b) to a curve, the full sequence uniquely determines both parameters. A geometric interpretation is provided using Thales semicircle partitions, where the associated altitude converges to its maximum at the fixed point. The results are presented as a mathematical exploration of contractive geometry in harmonic averaging, without asserting any physical interpretation.