A boundary that actually falls inside the date
Taken at noon in every year from 1930 to 2035, this date has put the Sun between roughly nine and ten and a half degrees of Gemini. Ten degrees is where the first decan ends and the second begins, so the sampled span lands on both sides of it.
That makes 31 May one of the few birthdays where the honest answer is that the decan depends on the year, and on the hour within that year. Anyone telling you flatly that this date is a first decan Gemini, or flatly that it is a second decan Gemini, is reading from a table that cannot see the difference. The Sun moves a little under a degree a day, so a birth in the early morning and one late at night can genuinely fall either side.
What changes when it crosses
The first face of Gemini is Jupiter's in the Chaldean order, which runs Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter from zero degrees of Aries. The second face is the next in that loop, Mars. Under the triplicity scheme the first belongs to Gemini itself and so to Mercury, and the second to Libra and so to Venus.
So a single degree of difference swaps Jupiter for Mars, or Mercury for Venus, depending on which scheme is in use. Tradition reads the earlier band as the wider or plainer expression of the sign and the later one as the sharper or more socially directed. Those are readings attached to a band, not statements about a person, and they say nothing about the sun sign itself, which is Gemini on either side.
Why the date does not wander further than it does
The whole variation here is about a degree and a half over more than a century, which is the ordinary amount for any date. Three common years each push the Sun's position for a given date forward by about six hours of motion, and the leap day pulls it back by a day, so the pattern repeats rather than drifting.
What makes this date interesting is not that it moves more than others but that a boundary happens to sit inside the window. The same is true of sign start dates, which is why the beginning of Gemini itself moves between 20 and 21 May while the middle of the sign looks fixed.
It also means the answer for one person born on 31 May tells you very little about the answer for another. The year and the hour both feed into it, and only a calculated position resolves the pair.
Settling it, and what stays unsettled
The widget above computes the Sun's longitude from a year, an hour and a birthplace, and reports the degree and minute, the face that degree falls in, and the distance to the nearest sign boundary. On this date the face is the part worth having computed, since it is the one thing a date range cannot deliver.
The rest of the chart is untouched by any of this. The Moon can be in one of two signs across the day and has been in all twelve over a run of years, and the rising sign depends on the hour and the latitude rather than on the date. Those remain open until a birth time is supplied.
Questions people ask about this
Is 31 May the first or second decan of Gemini?
It can be either. The noon positions across the last century run from about nine to about ten and a half degrees, and the boundary is at ten, so the year and the hour decide it.
Is 31 May still definitely Gemini?
Yes. All three decans lie inside Gemini, and the sign boundaries are weeks away in both directions. Only the face is in question here, never the sign.
How do I find out which face applies to me?
Use the widget above with your birth year, time and place. It computes the Sun's exact longitude rather than reading a range, which is the only way to resolve a ten degree band.
Which rulers are involved either side of the line?
Jupiter then Mars under the Chaldean order, or Mercury then Venus under the triplicity scheme. The two schemes disagree with each other more than the two faces disagree between themselves.
Does a birth time matter more on this date?
For the decan, yes. The Sun advances a little under a degree in a day, which is enough to cross the boundary in a year where the noon position is already near ten.
What if my birth time is unknown?
The calculation defaults to noon, which on this date can put you on the wrong side of the face boundary. If the result comes back within a degree of ten, treat the decan as unresolved.