Where the disagreement starts
Ten degree divisions of the signs are old, and the two surviving ways of assigning rulers to them do not line up. The Chaldean order is a plain repeating sequence, Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter, begun at zero degrees of Aries and carried around all thirty six faces without restarting at each sign. Counted through, the first face of Gemini falls on Jupiter.
The triplicity scheme ignores that sequence. It gives the decans of a sign to the signs of the same element in order, so the air decans of Gemini go to Gemini, Libra and Aquarius. The first is Gemini's own, which means Mercury. So a 26 May birth is a Jupiter decan or a Mercury decan depending on which book is open, and the honest answer is to name the scheme rather than pick a winner.
The Sun's position on the date
Taken at noon across 1930 to 2035, a 26 May Sun has stood between about four and six degrees of Gemini. It never leaves the first decan and it never approaches the sign boundary, which is why the date is so stable.
The narrowness comes from the calendar rather than from anything astronomical about the date. The leap year cycle keeps the civil calendar and the orbit in step to within about a day, so the same date returns to nearly the same point on the ecliptic year after year.
Within a single year the Sun still moves a little under a degree a day, so a birth just after midnight and one just before the next midnight differ by that much. On this date the difference changes nothing about the sign or the decan.
What tradition attaches to this band
Under the Chaldean reading, Jupiter over an air sign is described as widening Gemini's usual quickness into something more discursive, more interested in the general case than the immediate detail. Under the triplicity reading the same band is Gemini doubled, Mercury over Mercury, and is described as the sign at its most concentrated.
Both of those are interpretation and should be read as such. The checkable claim underneath them is only that the Sun stood in the first ten degrees of Gemini, and that claim can be verified for any birth with a year, an hour and a place.
The parts the date says nothing about
Nothing about 26 May fixes the Moon. It moves fast enough to change sign roughly every two and a half days, so on this date it occupies one sign or two, and over a run of years it has been in all of them.
The rising sign is decided by the hour and the location, not the date, and every one of the twelve is possible. The widget above takes those details and returns the Sun's exact degree along with its distance from the nearest boundary, which on this date is a wide margin rather than a close call.
If the hour of birth is unknown the calculation falls back to noon. For the Sun on 26 May that costs almost nothing, since the entire day sits in one sign and one face, but any Moon or Ascendant produced from a noon default should be treated as unconfirmed.
Questions people ask about this
Which decan ruler applies to 26 May?
Jupiter under the Chaldean order and Mercury under the triplicity scheme, since both describe the same first decan of Gemini. Neither is more correct, but a writer who does not say which they are using is worth reading carefully.
How far into Gemini is 26 May?
About four to six degrees, so a fifth of the way through the sign at most. The exact figure depends on the year and the hour of birth, and the widget above will give it.
Could a 26 May birth ever be Taurus?
Not at noon in any year from 1930 to 2035. The Sun leaves Taurus around 20 or 21 May, several days before this date, so the margin is comfortable rather than borderline.
Does the decan change the sun sign?
No. A decan is a subdivision inside a sign, not a rival to it. The Sun is in Gemini for the whole of all three decans, and the face only adds a second layer of description.
Is this a cusp birthday?
No. It sits well inside Gemini and there is no boundary near it. Cusp language usually signals missing information rather than a genuine mixed position, and here there is nothing missing.
What should I look up next if the sign is already settled?
The Moon and the rising sign, since those are the placements the date leaves open. Both need a birth time, and the rising sign also needs a birthplace.