= Solution
Take forward as increasing $x$, and specify the sinusoidal amplitude convention by
$$
U(x)=A[1-\cos(2\pi x/d)]-Fx.
$$
Thus the unforced barrier is $2A$, not $A$. Let $s=Fd/(2\pi A)=0.7051$ and $\theta=\arcsin s$. Stationary points exist because $F<2\pi A/d=7.09\,\mathrm{pN}$. A minimum is at $x_m=d\theta/(2\pi)$ and its forward saddle at $x_+=d(\pi-\theta)/(2\pi)$. The distances and activation barriers are
$$
\ell_+=\frac d2-\frac{d\theta}\pi=1.756\,\mathrm{nm},\quad
\ell_-=d-\ell_+=5.244\,\mathrm{nm},
$$
$$
\Delta U_+=2A\sqrt{1-s^2}-F\ell_+=2.422\,\mathrm{pN\,nm},\qquad
\Delta U_-=2A\sqrt{1-s^2}+F\ell_-=37.422\,\mathrm{pN\,nm}.
$$
Use an <Arrhenius equation>[Arrhenius estimate] with the supplied distance-squared waiting-time scaling, $\tau\propto\ell^2e^{\Delta U/(k_BT)}$. Calibrate at $F=0$ with $\ell_0=d/2$ and rate $k_0=1\,\mathrm{s^{-1}}$ in each direction. At the explicit room-<temperature> estimate $T=300\,\mathrm K$, $k_BT=4.142\,\mathrm{pN\,nm}$, giving
$$
\boxed{k_+\simeq k_0\left(\frac{d/2}{\ell_+}\right)^2e^{(2A-\Delta U_+)/(k_BT)}\simeq100\,\mathrm{s^{-1}}.}
$$
Applying the same distance rule separately backwards gives $k_-\simeq0.00241\,\mathrm{s^{-1}}$ and hence the literal scaling-model estimate
$$
\boxed{f_-^{\rm distance\ rule}=\frac{k_-}{k_++k_-}\simeq2.4\times10^{-5}.}
$$
\b[Thermodynamic qualification:] for isothermal <diffusion> in this translated tilted potential, adjacent wells differ by $Fd=35\,\mathrm{pN\,nm}$, so <local detailed balance> instead requires $k_+/k_-=e^{Fd/(k_BT)}$. The two separate distance prefactors introduce an extra factor $(\ell_-/\ell_+)^2\simeq8.91$, inconsistent with that requirement. Keeping the forward estimate while enforcing the proper ratio gives
$$
\boxed{k_-^{\rm DB}\simeq0.0215\,\mathrm{s^{-1}},\qquad
f_-^{\rm DB}=\frac1{1+e^{Fd/(k_BT)}}\simeq2.14\times10^{-4}.}
$$
Both interpretations are shown because the supplied distance heuristic is not, by itself, a thermodynamically consistent prescription for both directional prefactors. Additional hidden chemical driving would require its own affinity, not an unexplained prefactor asymmetry. A common transition-state prefactor, as in a consistent <Kramers escape rate>[Kramers model], preserves the <local detailed balance>[Boltzmann rate ratio]. Moreover the forward barrier is only about $0.58k_BT$, so rare-escape exponential dwell kinetics and the numerical rate should be regarded as rough estimates. If “amplitude” meant the entire unforced barrier height, the maximum restoring <force> would be only $\pi A/d=3.55\,\mathrm{pN}$ and no dwell wells would survive at $5\,\mathrm{pN}$; the amplitude convention is therefore essential.
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