Frequencies (ticking rates) of the two ${}^{87}\text{Sr}$ clocks in the experiments at the Tokyo Skytree broadcasting tower In 2019, researches in Japan (M. Takamoto, H. Katori et al.) carried out experiments comparing two transportable ${}^{87}\text{Sr}$ clocks, one ("clock $\mathsf 1$") placed at the ground floor and the other ("clock $\mathsf 2$") placed in the observatory floor of the Tokyo Skytree broadcasting tower, nominally $450~{\rm m}$ above the ground floor. There's an article filed, with a public supplement and with a public article copy available.
The article introduces the symbols $\nu_{\mathsf 1}$ and $\nu_{\mathsf 2}$ to denote "the clock frequency at location $\mathsf 1$" and "the clock frequency at location $\mathsf 2$", respectively; and also referred to as "the clock laser frequencies".
Ramsey spectra (Excitation probability over Detuning) measured in the observatory floor and the ground floor with fit curves are shown (Fig. 2 b,c) with the maximum peak for clock $\mathsf 2$ at the reported Detuning value $\approx 21.18~{\rm Hz}$ and the maximum peak for clock $\mathsf 1$ apparently at Detuning value $0~{\rm Hz}$. (For reference, let's denote these two quantities as $f^{RSmax}_{\mathsf 2 \leftarrow\mathsf 1}$ and $f^{RSmax}_{\mathsf 1 \leftarrow\mathsf 1}$, resp.)
This is surely a quite significant finding, considering that

*

*the full width at half maximum of the maximum peak of either Ramsey spectrum is apparently also in the order of $\approx 20~{\rm Hz}$,


*the CIPM recommended frequency of the relevant unperturbed ${}^1 S_0 - {}^3 P_0$ optical transition of ${}^{87}\text{Sr}$ is given with accuracy better than $1~{\rm Hz}$, as $f({}^{87}\text{Sr}) = 429 \, \, 228 \, \, 004 \, \, 229 \, \, 873.7 (0.5)~{\rm Hz}$, and


*the experiments involve corrections of about $+2.6~{\rm Hz}$, and systematic uncertainties much less than $\pm 0.1~{\rm Hz}$; as listed in Table S1 of the supplement.

Now, from the outset, in its abstract, the article claims that "A clock at a higher altitude ticks faster than one at a lower altitude, in accordance with Einstein’s theory of general relativity";
and it goes on to refer to "frequency shift $\Delta \nu = \nu_{\mathsf 2} - \nu_{\mathsf 1} \approx 21.18~{\rm Hz}$".
Therefore
My question:
Is it correct to conclude that clock $\mathsf 2$, while located at the observatory floor of the Tokyo Skytree broadcasting tower, ticked significantly faster than clock $\mathsf 1$, while located at the ground floor; in particular that (rounding down to integer ${\rm Hz}$)

*

*the clock frequency at location $\mathsf 2$ had the value $\nu_{\mathsf 2} \approx 429 \, \, 228 \, \, 004 \, \, 229 \, \, 897~{\rm Hz}$ while


*the clock frequency at location $\mathsf 1$ had the value $\nu_{\mathsf 1} \approx 429 \, \,  228 \, \, 004 \, \, 229 \, \, 876~{\rm Hz}$
?
(Or, to consider at least one alternative explicitly:
Is it instead correct to conclude that both clocks ticked as good as equally fast, both, at their respective locations, at approximately $\nu_2 \approx \nu_1 \approx 429 \, \, 228 \, \, 004 \, \, 229 \, \, 876~{\rm Hz}$; but the reported "frequency shift" value is instead attributable to the (suitably signed) difference between the tick frequency of a clock and the frequency of a receiver in response to ticks of that clock:
$$-(\nu_{\mathsf 2} - \nu^{\text{rec}}_{\mathsf 1 \, \leftarrow \, \mathsf 2}) \approx (\nu_{\mathsf 1} - \nu^{\text{rec}}_{\mathsf 2 \, \leftarrow \, \mathsf 1}) \approx 21.18~{\rm Hz}$$
?)
 A: 
Is it correct to conclude that clock , while located at the observatory floor of the Tokyo Skytree broadcasting tower, ticked significantly faster than clock , while located at the ground floor


Is it instead correct to conclude that both clocks ticked as good as equally fast, both, at their respective locations … but the reported "frequency shift" value is instead attributable to the (suitably signed) difference between the tick frequency of a clock and the frequency of a receiver in response to ticks of that clock

The distinction you are drawing is a distinction without a difference. It is only a matter of your arbitrary choice of reference frame. The first statement would correspond to a non-inertial reference frame where both clocks are at rest. The second would correspond to an inertial reference frame where both clocks are accelerating.
The choice between the two descriptions is completely arbitrary and makes no real difference. The authors apparently were primarily using the first description, but if you prefer the second then you are free to re-cast their results in such terms. Neither is inherently right or wrong. Either is acceptable and is justified both experimentally and theoretically.
