= Fresnel convolution of free-particle time slices
{c}
{title2=$K_{t_1}*K_{t_2}=K_{t_1+t_2}$}
Completing the square in the <integral> over an intermediate <position> shows that two <free-particle propagators> compose to the <propagator> for the sum of their positive time intervals. For equal slices, the quadratic identity is
$$
\frac{(x-x_0)^2}{k}+(x_1-x)^2
=\frac{k+1}{k}\left(x-\frac{x_0+kx_1}{k+1}\right)^2
+\frac{(x_1-x_0)^2}{k+1}.
$$
The <Fresnel integral> contributes a factor proportional to $\sqrt{k/(k+1)}$; multiplying successive factors gives $1/\sqrt N$. A consistent convergence regulator fixes the phases of these square roots.
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