How to know the true value of the measured quantity from an interferometer phase? For an interferometer, the measured signal will oscillate as a function of the accumulated phase $\phi(x)$ as a sinusoid or cosine, where $x$ is a quantity we are trying to measure from the interference. The signal will have the same properties for all $\phi(x_2)=\phi(x_1)+2n\pi$ where n is an integer, my question regards how would we know from the signal, whether the quantity of interest $x$, is $x_1$ or $x_2$ considering we don't know $n$. Also, how would we be able to differentiate between negative and positive quantities of $x$?
For example, if the signal is due to a rotation $x\rightarrow\Omega$ of the frame of reference and the effect causing the phase accumulation the Sagnac effect $\phi(\Omega)=A\Omega$, where $A$ is just some constant we need not concern ourselves with for now. Then the signal output from the interferometer might go something like $S(\Omega)=C\cos(A\Omega)$, where $C$ is the contrast. Clearly, the signal is the same for $\pm\Omega$ and we can't tell the difference between large $\Omega$ and small $\Omega$ if the amplitude of the signal is the same for those two values of $\Omega$.
 A: This is a great example why physical theories are necessary a priori for interpretting the results of any experiment. One must first have an idea of the magnitude of the effect they are trying to measure before really being able to ascertain its value. When we are talking about a physical rotation, perhaps all that matters is the final orientation, so the extra $2\pi n$ may not matter. When we are talking about a relative phase between two oscillators, the extra $2\pi n$ does not matter.
But when we are talking about a cumulative effect, the total phase matters! So, if you want to measure a cumulative effect using an interferometer, you must already know within which $2\pi$ window you expect to find anything... otherwise you'll be out of luck. For something like gravitational wave detection, we might know that the signal is always rather small, so we might only expect to use $n=0$. For other cases, without a priori assumptions or knowledge, you are absolutely correct that an interferometer will only give you a definite answer modulo factors of $2\pi$.
