= Solution
Use <canonical quantization> with $[\hat q,\hat p]=i\hbar$, $\hat p=-i\hbar\partial_q$, and $\hat H=\hat p^2/(2m)+V(\hat q)$. Assume a real <potential energy> with the regularity and lower-bound conditions needed for a self-adjoint <Hamiltonian> and its <Trotter product formula>; singular potentials require their domain and limiting prescription to be specified separately. The transition amplitude is $K(\beta,\alpha;T)=\langle\beta|e^{-iT\hat H/\hbar}|\alpha\rangle$.
Take $\delta=T/N$, $q_0=\alpha$, $q_N=\beta$, and choose the left-endpoint <potential energy> convention. The precise time-sliced <phase-space path integral> is
$$
K_N=\int\prod_{j=1}^{N-1}dq_j\prod_{j=1}^{N}\frac{dp_j}{2\pi\hbar}
\exp\left\{\frac{i}{\hbar}\sum_{j=1}^{N}\left[p_j(q_j-q_{j-1})-\delta\left(\frac{p_j^2}{2m}+V(q_{j-1})\right)\right]\right\}.
$$
The endpoint positions are fixed and the momenta are unconstrained. Its continuum notation is
$$
\boxed{K=\int_{q(0)=\alpha}^{q(T)=\beta}\mathcal Dq\,\mathcal Dp\;
\exp\left[\frac{i}{\hbar}\int_0^T(p\dot q-H(p,q))dt\right],}
$$
where the functional measure means the displayed limit, not an unspecified product of unnormalized measures.
At each slice, complete the square in <momentum>. With the square-root branch selected by continuation from positive Euclidean time,
$$
\int\frac{dp}{2\pi\hbar}\exp\left[\frac{i}{\hbar}\left(p\Delta q-\frac{\delta p^2}{2m}\right)\right]
=\left(\frac{m}{2\pi i\hbar\delta}\right)^{1/2}
\exp\left[\frac{im(\Delta q)^2}{2\hbar\delta}\right].
$$
Thus the <time-sliced equivalence of phase-space and configuration-space path integrals> holds already at finite $N$:
$$
K_N=\left(\frac{m}{2\pi i\hbar\delta}\right)^{N/2}
\int\prod_{j=1}^{N-1}dq_j\;
\exp\left\{\frac{i}{\hbar}\sum_{j=1}^{N}\left[\frac{m(q_j-q_{j-1})^2}{2\delta}-\delta V(q_{j-1})\right]\right\}.
$$
This defines the <configuration-space path integral>
$$
\boxed{K=\int_{q(0)=\alpha}^{q(T)=\beta}\mathcal Dq\;
\exp\left[\frac{i}{\hbar}\int_0^T\left(\frac m2\dot q^2-V(q)\right)dt\right].}
$$
There are $N$ Gaussian prefactors but only $N-1$ coordinate integrations. Omitting that distinction would change the kernel normalization. The paths integrated in the limit need not be differentiable; the kinetic action notation stands for the lattice difference expression.
To identify this with the operator amplitude, normalize $\langle q|p\rangle=(2\pi\hbar)^{-1/2}e^{ipq/\hbar}$. The kernel of one product factor is exactly
$$
\langle q_j|e^{-i\delta\hat p^2/(2m\hbar)}e^{-i\delta V(\hat q)/\hbar}|q_{j-1}\rangle
=\int\frac{dp_j}{2\pi\hbar}\;e^{ip_j(q_j-q_{j-1})/\hbar-i\delta p_j^2/(2m\hbar)-i\delta V(q_{j-1})/\hbar}.
$$
Insert $N-1$ position resolutions of the identity between factors. Their product is precisely $K_N$. The <Trotter product formula> then gives
$$
\left(e^{-iT\hat p^2/(2mN\hbar)}e^{-iTV(\hat q)/(N\hbar)}\right)^N\longrightarrow e^{-iT\hat H/\hbar},
$$
so both functional integrals represent the same canonical kernel. One can define the finite integrals first at $T=-i\tau$, $\tau>0$, where the Gaussian <momentum> and coordinate kernels converge, and then continue to the real-time boundary value. Alternatively use an equivalent oscillatory damping prescription. Operator convergence fixes the distributional kernel limit; it need not imply pointwise convergence at every endpoint for every admissible <potential energy>. There is no operator-ordering ambiguity for this separated kinetic-plus-potential <Hamiltonian> once the common slicing convention is fixed.
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