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Ed,

I'm not entirely sure what you're saying here, but there are a couple of tricks to identify Greenwich Time in these calculations when you're unsure. Suppose I have a pair of calculations on a bit of scrap paper which I believe to be calculations of LMT and GMT. You might see something like:

8:13:30
   +4:13
-----------
8:17:43

and
15:10:45
    -7:35
-----------
15:03:10

How can you tell which of these is the GMT calculation? They look alike superficially. If there's no longitude written down, you might not know which is which...

First clue: look for something on the page indicating the ship time when the sight was taken. In those scrap paper sights which we have been calling the "Elizabeth sights" in this thread (about one-third to half of the work was done by Elizabeth Layton in this notebook, about half by John Layton, master of the vessel, and some fraction by the first mate), very often at the top of the work, the time is given, e.g. "3:15 P.M." This is time off their ordinary watches, and it would generally have been determined by the setting of the watches at the previous noon, either that date or the previous day. When the time sight is worked up, it will give a time reasonably close to this, but because setting watches at noon is inaccurate (due to the hang time around noon) and because the vessel has travelled and because pocket watches are not necessarily accurate, the time observed in the sight will differ by some minutes. So if the work begins with "3:15 P.M." and we find a time of "15:10:45" (as above) in the work, then that's a safe bet that this is the calculated local time.

By the way, that knowledge of approximate local time when starting into the math of a time sight saves a fair bit of digging time in the tables. There's that point in the calculation when you're "coming out of" logarithms... doing a reverse lookup in those long columns of logs to pull out a time. The tables had time columns (A.M. and P.M.) so if you know it's about 3:15 in the afternoon, you start your search for your calculated logarithm there! This eliminates most of the page-flipping that you would see in a textbook or classroom exercise of work like this. Another way of saying this: we already know what time it is -- the sextant sight and the calculation for local time is merely giving us a refined value.

Second clue: the correction added or subtracted to the time will be either explicable, as the equation of time, or completely inexplicable. In my example above, if we know from other information that it is around September 23, then we know that the equation of time will be about 7m30s and that the Sun is fast at that time, so the equation of time would be a subtraction of a value close to that. This is what I mean by an "explicable" value. We can see where it comes from. So when see 7:35 subtracted from 15:10:45, we once again can see that this must be local time. On the other hand, for GMT, it was common to write down the direct reading from the chronometer, like 8:13:30 in my example above, and then add some correction to it. At first glance, this correction looks an awful lot like the equation of time correction, but there's a big difference: it has no explanation. We can't go beyond that number as written. We have to take it a face value since it is simply a property pulled from a table that they are maintaining somewhere else (probably kept with the chronometer itself) that shows the accumulated error for any date based on the best estimate of the daily error determined the last time the chronometer was rated (probably the last time they were in their primary port for more than a few weeks). If the number is inexplicable, it's probably chronometer error.

If you were wondering about the twelve hour jumps that can occur when we go from sea days (also known as nautical days) to civil days or astronomical days, yes, those can be tricky, but as long as there's some piece of astronomical data written down from an almanac, it's normally possible to deal with that. This is especially true with worked lunars. The data for lunar distances never repeats. If I know the date within some margin of error, it's possible to figure out the true hour, day, and year (in the modern calendar, in normal civil timekeeping), just by looking at a listed lunar distance angle. It's also possible to do this with the declination of the Sun though with less certainty, and of course that doesn't work well near the solstices since the Sun's Dec changes slowly then.

Frank Reed