= Solution
Put $\xi_t=\ker\alpha_t$, the <contact distribution>. If $\dim M=2r+1$, the <contact form> condition is $\alpha_t\wedge(d\alpha_t)^r\ne0$. It implies that $d\alpha_t|_{\xi_t}$ is a <symplectic form> on each contact hyperplane: inserting a vector transverse to $\xi_t$ in that volume form leaves the nonzero top exterior power $(d\alpha_t|_{\xi_t})^r$. Hence there is a unique smooth <time-dependent vector field> $Y_t\in\xi_t$ satisfying
$$
(\iota_{Y_t}d\alpha_t)|_{\xi_t}=-\dot\alpha_t|_{\xi_t}.
$$
Its smoothness follows by inverting the smoothly varying <nondegenerate> matrix of this <contact distribution symplectic form>. This is the <horizontal generator of contact stability>.
Let $R_t$ be the <Reeb vector field> of $\alpha_t$. The one-form $\dot\alpha_t+\iota_{Y_t}d\alpha_t$ vanishes on $\xi_t$, so it is $h_t\alpha_t$. Evaluating on $R_t$, using $\alpha_t(R_t)=1$ and $\iota_{R_t}d\alpha_t=0$, identifies the coefficient:
$$
h_t=\dot\alpha_t(R_t),\qquad
\dot\alpha_t+\iota_{Y_t}d\alpha_t=h_t\alpha_t.
$$
By <Cartan's magic formula>, the <Lie derivative of a differential form> is
$$
\mathcal L_{Y_t}\alpha_t=\iota_{Y_t}d\alpha_t+d(\alpha_t(Y_t))
=\iota_{Y_t}d\alpha_t,
$$
because $Y_t$ lies in the contact hyperplane.
Let $\rho_t$ solve $\partial_t\rho_t=Y_t\circ\rho_t$, with $\rho_0=\operatorname{id}_M$. Since $M$ is compact without boundary and $Y_t$ is smooth on the closed time interval, its solutions cannot escape and exist for the entire interval. Reversing the time-dependent equation supplies a smooth inverse for each $\rho_t$. Thus these <flow maps> give a <smooth isotopy>. Differentiating the <pullback of a differential form> along this isotopy yields
$$
\frac d{dt}(\rho_t^*\alpha_t)
=\rho_t^*(\dot\alpha_t+\mathcal L_{Y_t}\alpha_t)
=(h_t\circ\rho_t)\rho_t^*\alpha_t.
$$
For each base point this is a scalar linear equation for a covector, initially $\alpha_0$. Its solution is
$$
\boxed{\rho_t^*\alpha_t=u_t\alpha_0,\qquad
u_t(x)=\exp\left(\int_0^t h_s(\rho_s(x))\,ds\right)>0.}
$$
By <finite-time flow completeness on a compact manifold>, this construction gives the whole time interval. The <contact conformal factor along an isotopy> is smooth in $(t,x)$ and nowhere zero, proving <Gray stability theorem>. The precise domains here are $\rho_t:M\to M$ and $\rho:M\times[0,1]\to M$; the extra time factor attached to the already indexed $\rho_t$ in the source is a notational slip. The proof produces the required family on its whole specified interval. If a parameter domain of all $\mathbb R$ is intended, extend the smooth field slightly beyond $[0,1]$, multiply it by a time cutoff, and use compactness to obtain the extended isotopy $\rho:M\times\mathbb R\to M$. Its restriction to $[0,1]$ is unchanged.
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