For an average adult, in general, if one knows they will wake up in the middle of a cycle--is it better to go to bed later? Let us say that one consistently goes to sleep at the same time every day, and sets an alarm for the same duration of sleep 7 hours and 45 minutes (e.g., 10:45pm to 6:30am). 

One day, they get caught up in something, and are 30 minutes late to "bedtime." Is it better to wait 45-60 minutes to wake up at the same time (at the end of a cycle), refreshed, and not in the middle of a cycle, groggy?



I know there are some odd assumptions here, but if you could please either help edit or have better questions (or advice) to strike more at the heart of the matter--please do so. I think the largest assumption is the roughly 90 minutes sleep phase cycles at the end at least:
As a precursor, even if a stringent sleep schedule is absolutely followed, one still needs to practice other good sleep hygiene habits: avoiding blue light (or using a filter for screens), exercising regularly, eating healthy and attaining an overall equanimous mood (lack of distress). That, among other non-ideal factors, such as sleep disturbances could also result in a poor nights sleep. But the question was stated as if in a 'perfect vacuum.' Non-idealities are overlooked.

That being said, I am going off the aforementioned assumptions about religiously going to bed at 10:45 and waking at 6:30 every day--including weekends. Let's say this person could sleep for nine hours (9:15 TIB), and we can safely assume that an adequate nights sleep is 7:45 TIB. In other words, this person can be high-functioning on 7.5 hours of sleep (five phases).

→ Is it better to wait until 12:15? (Either way, 6:30 wakeup for work stays constant.) One deviant day--does it matter? When will the sleep debt be relinquished? Honestly, I didn't find any research remotely about it.

Let's go out on a limb, and say it doesn't matter given the ideal-vacuum conditions. Then how many deviations, say, twice a week--do matter? How many will lead to partial sleep deprivation? Still, the answer is unknown. 



So I emailed sleep researchers from U.W. Madison and Stanford: 

Going to sleep in the middle of a phase versus "catching" the next phase (with less sleep) for one night (subject)


  Hello Professor _______,
  
  I have question for you about making a bedtime choice, a deviation,
  from a consistent sleep schedule: For an average adult, in general, if
  one knows they will wake up in the middle of a cycle--is it better to
  go to bed later?
  
  Scenario1
  
  Get home late, and ready to be in bed at 11:15 PM. Must wake at 6:30
  for work.
  
  Scenario2
  
  Get home late, later, and ready to be in bed at 11:45 PM. Must wake at
  6:30 for work.
  
  The first is 30 minutes late. The second is 60. Let's say for this
  person that we can safely assume that an adequate nights sleep is 7:45
  TIB. In other words, this person can be high-functioning on exactly
  7.5 hours of sleep (while following a consistent sleep/wake routine).
  
  → Is it better to wait until 12:15?
  
  [...] I ask this question because I follow the routine above. I
  deviate 1-2 times a month, and either way usually take a nap before
  3PM to follow, but persistently wonder which is the ideal choice.




Prof. Responses


  Hi Adam, 
  
  There is not easy answer, I do not know of direct evidence
  for one schedule over the other;  you are the best judge;  try the
  different schedules and  decide which one works for you (i.e. you feel
  better and you are “ready” to go) 
  
  It may be that they are very
  similar; in any case, it is dangerous to assume 7 or 8 hours is what
  YOU  need 
  
  Inter individual variability is huge, this is why each
  person is the best judge


~ C.C.


  Adam,
  
  Unfortunately, there is no good answer. The data for waking up in the
  middle of a cycle is sparse and doesn't offer good conclusions (the
  only reasonable one is that if it makes you feel better to wake up at
  the end of a cycle, that's probably a good thing). There are some
  circadian consequences of bouncing around a sleep onset time too much,
  with greater variability in onset time leading to more difficulty
  initiating sleep (and probably some other changes as well, though less
  well documented). Me, I'd stick with the earlier bed time that is
  closer to standard bedtime and get the extra sleep...


~ J.Z.

Going Further

Assess your ideal amount of sleep by taking a 'sleep vacation.' Over a two-week period, keep your consistent bedtime, but don't wake up with an alarm clock. Log. (See healthysleep.med.harvard.edu/need-sleep/what-can-you-do/assess-needs.)

How do you objectively analyze your sleep?


  There are two commonly used methods to assess sleep, Actigraphy
  and Polysomnography. Actigraphy is a non-invasive method of
  monitoring sleep and involves wearing wrist activity monitors, which
  are devices worn like a wristwatch that continuously record body
  movement (usually stored in 1-min epochs). [...] Actigraphy is useful for
  understanding sleep patterns as it is non-invasive and relatively
  easy to collect data over significant periods of time (commonly two
  weeks of monitoring).
  
  The second method is Polysomnography (PSG) which is a sleep
  study in which body functions such as EEG, eye movements,
  muscle activity and cardiac activity are measured. PSG provides
  information on sleep staging and is considered the “gold standard”
  for assessing sleep quality and quantity.


~ Sports Science Exchange (2013) Vol. 26, No. 113, page 2

→ DIY: https://sleepjunkies.com/features/the-ultimate-guide-to-sleep-tracking



Additional Resources


Phase Relationships between Sleep-Wake Cycle and Underlying Circadian Rhythms in Morningness-Eveningness
Modeling the Effects of Sleep Debt: On the Relevance of Inter-Individual Differences


Extra


https://faculty.ist.psu.edu/vhonavar/Papers/BIBM2017_YE.pdf