An optical double is two stars which appear very close together because they chance to be lined up as seen from Earth but are actually tens, hundreds or thousands of light years apart in space.
A real double star is a binary, where the two stars are gravitationally bound and orbit around each other.
A naked eye binary star is a binary were both of the stars are bright enough to be seen with the naked eye from each and where the physical separation between them is far enough enough for them to be seen from Earth as two separate stars by the unaided human eye instead of being blurred together into a single dot of light. Seeing the two stars is called "splitting" the binary star.
In my opinion there are no binary stars which can be "split" with the naked eye that have orbital periods less than 100,000 years.
Stars I have checked for orbital periods include Mizar and Alcor, which may or may not be gravitationally bound and I haven't found a period listed for them; Epsilon Lyrae with a period of hundreds of thousands of year; Alpha Librae with a period over 200,000 years, Beta Capricornii 700,000 years; and Zeta Corvi and HR 4691 that may or may not be a binary with a period of about 3,500,000 years.
So I am doubtful that any naked eye binary stars have orbital periods less than 100,000 years.
So does anyone know of any naked eyes binary stars with orbital periods less than 100,000 years?