Where the Sun stands
Measured at noon, the solar longitude on 4 September has run between about 11.0 and 12.4 degrees of Virgo in every year from 1930 to 2035. The twelfth degree is the working answer, and the whole spread is narrow because the leap year cycle keeps a fixed calendar date returning to nearly the same point in the orbit.
That places the Sun comfortably inside Virgo's middle face, which runs from ten degrees to twenty. Neither edge is anywhere close, so unlike the first days of the month this date does not need the year checked before the face can be named.
The Chaldean count
The older of the two schemes assigns the thirty six faces of the zodiac to the seven traditional planets in the order Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter. The loop starts at the beginning of Aries and simply repeats, so it takes no notice of sign boundaries and each sign's three faces always draw three consecutive planets from the sequence.
Counting through, Virgo's faces come out as the Sun, then Venus, then Mercury. This date falls on Venus. Tradition reads a Venus face in an earth sign as care about form as well as function, an eye for proportion attached to Virgo's habit of correcting things.
The triplicity count
The competing scheme is organised by element. Each sign's three faces go to the three signs of its own element, taken in zodiacal order and starting with the sign itself. The earth signs run Taurus, Virgo, Capricorn, so Virgo's faces belong to Virgo, then Capricorn, then Taurus.
By that count this birthday falls under Capricorn, and so under Saturn. The reading turns towards endurance, structure and long horizons rather than towards proportion. There is no procedure for choosing between the two systems, and the practical response is to notice which one a given description assumes before taking anything from it.
The two counts also behave differently at sign boundaries. The Chaldean loop runs straight through them, so the last face of one sign and the first of the next take consecutive planets, while the triplicity scheme restarts with each sign and always gives a sign its own first face. That structural difference is the reason the schemes diverge as often as they do.
What neither scheme touches
Both are layers on the Sun, and the Sun is the placement the date already gives away. Everyone born on 4 September in a given year shares it. The Moon is narrowed to two possible signs by the date and pinned by the hour, since it crosses about thirteen degrees of the zodiac a day.
The Ascendant is set by the Earth's rotation and can be any of the twelve signs, shifting roughly every two hours and by a degree for every four minutes of clock. Every house cusp follows from it. The decan question is worth understanding, but it is a detail beside the placements that need a birth time, and the widget above takes both.
Knowing which scheme is in play costs nothing and settles most of the apparent contradiction.
Questions people ask about this
What decan is a 4 September birthday?
The middle face of Virgo, running from ten to twenty degrees, with the Sun near twelve. Both edges of the face are far enough away that the year does not change the answer.
Is my decan ruler Venus or Saturn?
Both, in the sense that each scheme gives a different one for the same degrees. The Chaldean order says Venus, the triplicity order says Saturn through Capricorn. Older texts more often mean the Chaldean answer.
Does that make decan readings unreliable?
It makes them ambiguous unless the scheme is stated. The band of degrees is precise, the rulership attached to it is a matter of tradition, and honest writing says which tradition it is following.
Is 4 September always Virgo?
Yes. The Sun is about twelve degrees into a thirty degree sign, so no year, hour or time zone brings it near either boundary.
How much does the position vary by year?
By a little over a degree in total across the last century, moving back and forth with the leap cycle rather than drifting in one direction. Your own value depends on the year, the hour and the birthplace.
What should I calculate next?
The Moon and the Ascendant. Both need a birth time, the Ascendant needs the birthplace too, and between them they account for most of what makes one chart differ from another on the same date.