The band and its two rulers
Gemini's middle decan runs from ten to twenty degrees, and a 2 June Sun sits in its opening stretch. The Chaldean order, which runs Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter in a loop from zero degrees of Aries without restarting at each sign, arrives at Mars for this face.
The triplicity scheme has nothing to do with that sequence. It gives a sign's decans to the signs of its own element in order, so the air decans of Gemini fall to Gemini, Libra and Aquarius. The middle one is Libra's, and Libra's ruler is Venus. Mars and Venus are opposite in almost every traditional description, which makes this a good band for seeing how much of decan writing depends on an unstated choice of scheme.
What tradition reads into each version
The Mars reading treats Gemini's usual quickness as sharpened, more argumentative, more inclined to press a point than to circulate around it. The Libra reading treats the same quickness as directed at other people, weighted towards exchange and agreement rather than contest.
Both are commentary on a ten degree band, not statements about anyone born in it, and either would be replaced by the other if the writer picked up a different book. The checkable part is only the position, and that can be verified from a birth year, hour and place.
It is also the layer that separates one Gemini birthday from another. Everyone born across the month shares the sign, while the face narrows the group to about ten days of the year.
How firmly the date holds its degree
Sampled at noon across 1930 to 2035, a 2 June Sun has stayed between about eleven and twelve and a half degrees of Gemini. The whole century fits in a window of roughly a degree and a half, comfortably inside the middle decan and nowhere near either sign boundary.
The reason is the leap year rule. A quarter day of surplus accumulates for three years, moving the position later, and the leap day clears it. Without that correction a birthday would slide steadily through the zodiac, and after a few centuries this date would not be Gemini at all.
The part a date can never fix
The Moon travels roughly thirteen degrees a day, so on any given 2 June it holds one sign or crosses into a second, and across different years it has occupied all twelve. The rising sign turns through a full circle every day and depends on latitude as well as clock time, so it too is entirely open.
The widget above computes the Sun's degree and minute from a real birth year, hour and place, names the decan it falls in, and reports the distance to the nearest sign boundary. On this date that margin is wide in both directions, which is the confirmation worth having rather than the surprise.
Where a birth time is missing the calculation falls back to noon. That is harmless for the Sun on this date, since the whole day stays in one sign and one face, and it is not harmless for anything that depends on the hour.
Questions people ask about this
Who rules the decan a 2 June Sun falls in?
Mars under the Chaldean order and Venus, by way of Libra, under the triplicity scheme. Both refer to the same middle decan of Gemini, so a description is only meaningful if it names the scheme it follows.
What degree of Gemini is 2 June?
Between about eleven and twelve and a half degrees at noon across the last century. A birth early or late in the day shifts that by less than a degree either way.
Is the sign ever in doubt on this date?
No. The Sun entered Gemini around twelve days earlier and does not reach Cancer for about three more weeks, so no year in the sampled span puts the sign in question.
Does the decan replace the sun sign?
No, it sits inside it. A decan is a ten degree subdivision, so a 2 June Sun is in Gemini and in Gemini's middle face at the same time, with the face adding detail rather than competing.
Why do sources disagree about my decan ruler?
Because two different traditions are in circulation and they were built on different principles. The Chaldean one is a fixed planetary sequence and the triplicity one follows the elements, and neither is a corrupted version of the other.
What should I calculate if the sign is settled?
The rising sign and the Moon, since those are what the date leaves open. Both need a birth time, and the rising sign needs a birthplace as well.