In the context of delay line measurements, 10 Hz - 10kHz from the carrier is near the carrier. You will not be able to measure this with a delay line that is a few hundred feet long.
I am going to go out on a limb and say you are wrong. There is no structural reason why the delay line (aka one oscillator) set up cannot measure phase noise arbitrarily close to the carrier. According to
Phase Noise and AM noise measurement in the Frequency Domain, the noise floor of the one oscillator set up is reduced compared to the two oscillator approach. The graph given in support of this shows the noise floor rising as the the Fourier frequency of the phase noise approaches that of the carrier. Unfortunately, the justification for this graph is another paper that I have not been able to acquire. So, there is no way to check the argument that led to that graph. However, the text makes no mention of a "structural problem" that leads to the result.
There are plenty of practical problems with measuring phase noise close to the carrier. However, these are not specific to the one oscillator set up. They apply equally to the two oscillator set up. I will describe them in a separate post.
So, on to the argument that the delay line/one oscillator measurement set up is not structurally deficient as a measurement technique. This argument follows your lead in assuming transmission lines are perfect (not lossy and linear) and it assumes all electronic circuits are perfect (e.g., filters have cutoff frequencies that are exact - they do not drop off over a range of frequencies). In this regard, the argument assumes a bandpass filter that passes only the carrier frequency and the carrier frequency plus 1 Hz. This filter is placed on the oscillator output before the signal enters one side of the mixer and the delay line. So, the signal presented to the double balanced mixer on both sides comprises a 2 Hz band limited to the carrier frequency and the carrier frequency plus 1 Hz. Since the delay line is perfect, the amplitudes of the generated and delayed signal are exactly equal.
To keep the argument simple, it is assumed that the generated signal is either at the carrier frequency or at the carrier frequency plus 1. No frequencies between these two are possible. Also, it is assumed that we are interested in SSB measurements. That is why the filter does not admit a band comprising the carrier frequency plus or minus 1.
Consider the output of the mixer. Represent the carrier frequency by f
c. There are 3 situations to consider:
1) Both the generated and delayed signal are at f
c. The mixer/phase detector will indicate in-phase.
2) Either the generated or delayed signal is at the carrier frequency and the other is at the f
c+1. The mixer/phase detector will indicate out-of-phase.
3) Both the generated and delayed signal are at f
c+1. The mixer/phase detector will indicate in-phase.
The spectral density associated with the 1 Hz phase noise will be proportional to the fraction of time the mixer is
(Changed 7-29-28) in situation 2 divided by the fraction of time the mixer is in either situation 1 or 3 (I thought about this later and now think the formula should be:) in either situation 2 or 3 divided by the fraction of time the mixer is in situation 1. (The exact constant of proportionality requires further thought - but I probably won't spend any time on it, since this is a conceptual argument, not a proposal for an actual measurement setup) I am completely aware that this setup framework is impossible to achieve in practice. However, it demonstrates that the one oscillator set up is
structurally capable of measuring phase noise as close to the carrier as one may wish.
If you disagree or find fault with this argument, I welcome you to provide a counter-argument or refutation. However, I am not interested in playing 20 questions with you. So, if you follow your recent habit of patronizating discourse, I probably will not respond.