Is ‘Gotra’ has scientific evidence and proof ?

yes and no. the gotras are calculated according the the descendants of various rishi munis.

but after a lot of mixed gotra marriages throught the generations, the offsprings of the gotras are not pure.

in a patriachial society, we consider gotras. but we are omitting the mother's role in bringing out the offspring. so  i

i would say that in gotra there shd be no scientific evidence.

my friend's son married  his father's brother's daughter/

first cousins and paternal cousins to boot.

but their children are healthy and good in studies.

Yes, Gotras do have a scientific bearing. You would be surprised to know that 99.5% of DNA is common in all humans but it is the variation in the sequence that holds the key to distinction in our genetic make-up. Even in western nations the genetic councilors use the pedigree chart of prospective couples to determine if there are any chances of unhealthy offsprings. In India fortunately we had the gotra system that helped ensure genetic health of the offspring.  Now let me explain--based on my understanding of Mendel's law-- as to why it is unhealthy to marry a near relative:-   Genes of near relatives are very similar approximating 100%. You would agree that every family is genetically  prone to certain diseases, say for example diabetes, hypertension, thalessamia, sickle cell anaemia. A child is a product of combination of genes of his father and mother.   Now suppose family has Gene 'A' that is 25% more prone to inflict a disease & if the both parents carry the same gene then the chances of offspring to be affected by the disease will certainly be higher than 25%, it may be 50% also.     Unfornutantely we are criticising our Gotra system (though the violence associated with it is certainly unteanable & deplorable) but other countries have woken up to it. For example there were reports that the Saudi government is planning to make blood test compulsory for all prospective marriage seekers. Gotras, unlike caste, is based on science. Different Gotras are considered lineal descendants of different Rishis such as Bhardwaj, vashist, Kashyup. Problem arose when Sub-Gotras came into being & that resulted in confusion as it became contentious as to individuals of which gotras could marry which one.            (Please note that the above is based on my own understanding of the subject)