Tight upper bound on $d_{GH}(S^1,S^{2k+1})$: GPT's short proof

We describe GPT-5.6 Sol Pro's short proof that $2\cdot d_{GH}(S^1,S^{2k+1}) \le \tfrac{2πk}{2k+1}$. This is the odd-dimensional case of a tight upper bound by Harrison and Jeffs on the Gromov--Hausdorff distance between the circle and spheres.