A position with nothing near it
Across 1930 to 2035, a 4 June Sun at noon has stood between about thirteen and fourteen and a half degrees of Gemini. The nearest sign boundary is roughly two weeks of solar travel away in either direction, and the nearest decan boundary, at twenty degrees, is around six days away.
Dates like this are the ordinary case and they get almost no attention, because there is no argument to report. What can be reported instead is the position itself, which is more than the sign and more than any date range can supply.
It also means the sign needs no birth time to confirm. Someone born on this date at any hour anywhere on Earth has a Gemini Sun, and the only question left is how many degrees into the sign it stood.
How the calendar keeps it there
The tropical year runs a little over three hundred and sixty five days, and the surplus quarter day is what leap years exist to absorb. Each common year the Sun reaches a given longitude about six hours later by the clock, three years of that accumulate to roughly eighteen hours, and the leap day pulls the whole thing back by twenty four.
The net effect is a sawtooth of about a degree and a half, repeating every four years, with no long term drift within a human lifetime. That is why someone born on 4 June in 1948 and someone born on 4 June in 2018 have Suns within a degree and a half of each other despite seventy years between them.
The middle face of Gemini
Ten to twenty degrees is Gemini's second decan, and thirteen degrees sits solidly inside it. The Chaldean order, Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter counted from zero degrees of Aries, gives that face to Mars. The triplicity scheme assigns the air decans to Gemini, Libra and Aquarius in order, which makes the middle one Libra's and so Venus's.
Tradition reads the Mars version as the sign's quickness given a competitive edge and the Libra version as the same quickness pointed at company and agreement. These are readings attached to a band rather than claims about a person, and they change with the scheme rather than with the birth.
The safest way to use them is to note both and say which one produced any given description. A source that gives a decan ruler without naming its scheme cannot be checked against anything.
What a settled Sun does not settle
Knowing the Sun to within a degree tells you nothing about the Moon, which moves roughly thirteen degrees a day and can be in either of two signs across a single 4 June. Over a run of years it has been in all twelve.
The rising sign is decided by the hour and the birthplace. It advances through a whole sign in about two hours and the rate varies with latitude, so two people born the same day in the same city an hour apart can have different Ascendants and a completely different arrangement of houses. The widget above resolves all of that once a time and a place are supplied.
Questions people ask about this
What sign is 4 June?
Gemini, at noon in every year from 1930 to 2035. The date sits near the middle of the sign, so no year in the span comes anywhere near a boundary.
What degree does the Sun reach on 4 June?
Between about thirteen and fourteen and a half degrees of Gemini, depending on the year and the hour of birth. The widget above narrows that to a single degree and minute.
Which decan applies?
The second decan of Gemini, ten to twenty degrees. The Chaldean order gives it to Mars and the triplicity scheme to Venus by way of Libra, and both descriptions are in wide circulation.
Is being in the middle of a sign meaningful?
It means the sun sign is unambiguous and does not need a birth time to confirm. Tradition attaches no special force to a middle position, though the exact midpoint at fifteen degrees does carry some older commentary.
Does a birth time change anything here?
It changes the Sun's degree by less than one, which is irrelevant to the sign. It changes the rising sign, the houses and sometimes the Moon sign completely.
Why quote a degree range rather than one number?
Because the exact figure depends on the year and the hour of birth. The range is what the date alone supports, and a calculation is what narrows it to a point.