If he swallows saliva after it came out to the lips, does that break the fast, and is there scholarly consensus on the issue?
Question

                Are all the madhhabs agreed about the ruling on swallowing saliva, and that if it had come out to the lips, then swallowing it invalidates the fast? I hope that you can tell me if there is any view that regards it as permissible to swallow saliva if it has come out onto the lips, so that I can follow it, because I suffer from waswasah, and I am looking for the easiest rulings. Please note also that I do not feel entirely convinced that if the saliva has come out to the lips, swallowing it invalidates the fast. I feel that this is not logical. When I woke up from sleep, I found some saliva on the outside of my mouth, and I swallowed it deliberately.
Praise be to Allah.If the fasting person deliberately swallows his saliva after it had come out to his lips, he breaks his fast according to the Shaafa‘is and Hanbalis, because in that case the saliva has become detached from its place, which is the inside of the mouth, so swallowing it is like swallowing anything else that is separate from the mouth.
An-Nawawi (may Allah have mercy on him) said regarding the swallowing of saliva that does not break the fast: That is if he swallows it from its place, which is inside mouth. But if it came out of his mouth, then he took it back with his tongue or otherwise, and swallowed it, he has broken his fast.
Our companions said: Even if it came out to the lips, and he took it back and swallowed it, he has broken his fast, because he has made a mistake by doing that, and because it was outside the place where it may be overlooked. Al-Mitwalli said: If it comes out on his lips, then he takes it back and swallows it, he has broken his fast.
End quote from al-Majmoo‘ (6/342).
Ibn Qudaamah (may Allah have mercy on him) said: If his saliva comes out onto his garment, between his fingers, or between his lips, then he takes it back and swallows it, or he swallows the saliva of another person, that breaks the fast, because he swallowed it from somewhere other than his own mouth, so it is the same as if he swallows something other than saliva.
End quote from al-Mughni (3/17).
The Hanafis are of the view that it does not break the fast unless the  saliva became detached from the mouth, then he put it in his mouth.
It says in Fath al-Qadeer (2/332): If his saliva came out of his mouth, then he put it back in and swallowed it:
If it had not become separated from his mouth, rather it was still connected to what was in his mouth, like a thread, then he swallowed it, that does not break the fast.
But if it had become separated, then he took it and put it back, that does break the fast, although no expiation is required of him.
If it collected in his mouth then he swallowed it, that is makrooh, but it does not break the fast. End quote.
It says in Majma‘ al-Anhar (1/246): If his saliva came out of his mouth, then he put it back in his mouth and swallowed it:
If it had not become separated from his mouth, rather it remained connected to what was in his mouth, like a thread, then he swallowed it, that does not break the fast.
If it had become separated, and he took it and put it back, that does break the fast, although no expiation is required of him.
If his saliva got on both lips when speaking and the like, and he swallowed it, that does not break the fast. End quote.
In al-Jawharah an-Nayyirah (1/140), it says:
If the saliva of a fasting person runs down onto his chin when he is sleeping or otherwise, and he swallows it before it becomes separated from the mouth, that does not break the fast. End quote.
We have not come across any view of the Maalikis concerning this matter.
There is no blame on you if you follow the view of the Hanafis, because of the waswasah that you are suffering from; this is a valid excuse.
However it is more prudent to avoid doing that deliberately, without being unduly strict or paying heed to waswaas.
 And Allah knows best.