Approaching the centre of the sign
Sampled at noon in every year from 1930 to 2035, a 5 June Sun has stood between about fourteen and fifteen and a half degrees of Gemini. Fifteen is the halfway point of the sign, so most years fall just short of it and the later years in the leap cycle reach or pass it.
Older material gives the middle degree of a sign some attention, treating it as the point where the sign's qualities are least mixed with what lies on either side. That is commentary rather than measurement, and it applies to the position, not to anyone born there.
The practical value of the figure is different. A degree is what aspects are measured against, what a progressed chart advances from, and what decides which ten degree band the Sun occupies.
Why the figure is so reliable
A degree and a half of spread across more than a century is the whole variation, and it is the leap year cycle at work. The surplus quarter day in the tropical year builds up over three common years, moving the position later, and February the twenty ninth resets it.
Because of that correction the calendar does not slide against the seasons, and a birthday keeps its place on the ecliptic. The only other thing that moves the Sun on a given date is the hour of birth, which shifts it by a little under a degree from one midnight to the next.
So a 5 June Sun can be quoted before the year is known, and narrowed to a minute of arc once it is. That is a rare thing to be able to say from a calendar date alone.
In the middle of the middle face
Gemini's second decan runs from ten to twenty degrees, so fifteen is its centre as well as the sign's. The Chaldean order of Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter, counted from zero degrees of Aries, puts Mars on this face. The triplicity scheme allots the air decans to Gemini, Libra and Aquarius in order, which makes the middle one Libra's and therefore Venus's.
Those two readings pull in opposite directions, which is worth knowing before taking either at face value. What is not in dispute is the position: a Gemini Sun near fifteen degrees, in the second of the sign's three faces, in every year of the sampled span.
The rest of the chart is untouched
None of this constrains the Moon, which covers around thirteen degrees a day and can be in one of two signs across a single 5 June, or in any of the twelve across different years. Nor does it constrain the Ascendant, which turns through the whole zodiac daily and depends on the birthplace as much as the clock.
The widget above takes a year, an hour and a place and returns the Sun's degree and minute, the face it falls in, and the distance to the nearest sign boundary. On this date that distance is close to the maximum possible, which is a useful thing to have stated rather than assumed.
Questions people ask about this
How close is 5 June to the middle of Gemini?
Within a degree or so. The noon positions across the last century run from about fourteen to fifteen and a half degrees, and fifteen is the exact centre of the sign.
Does the midpoint of a sign mean anything?
Older sources treat the middle degrees as the least mixed part of a sign, away from the influence of what borders it. That is a traditional reading of a position rather than a claim about a person.
Which decan is a 5 June birthday?
The second decan of Gemini, from ten to twenty degrees, in every year sampled. Mars rules it in the Chaldean order and Venus in the triplicity scheme, by way of Libra.
Could a 5 June birth be Cancer?
No. The Sun does not reach Cancer until around 20 or 21 June, so there are more than two weeks of margin and no year in the sampled span alters that.
What does the year of birth change?
Only the exact degree, by up to a degree and a half, because of the leap year cycle. It changes neither the sign nor the decan on this date.
What still needs calculating after this?
The Moon sign in years where it changes during the day, the rising sign, and the house placements of every planet. All of those need a birth time, and the last two need a birthplace.