I noticed that the only official reason given for awarding Edward Witten the Fields medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully rigorous.

However, I came across this paper https://projecteuclid.org/download/pdf_1/euclid.cmp/1103921154 by Taubes and Parker which claims to make Witten's proof 'mathematically rigorous' and to justify assumptions which Witten made about Dirac operators. Does this mean that the Witten proof is not rigorous, or is it just the case that there were some unjustified lemmas to clear up which do not affect the validity or rigour of the argument (similar to the case of Perelman's proof of the Poincaré conjecture, where some lemmas and slight gaps had to be filled in)?

I am just curious as I have never really heard of the Taubes-Parker paper so I was assuming that the Witten paper was fully rigorous.

Later: I realise now that this is a bit of a misleading question, as 'proofs' can in reality come in many different flavours and/or levels of rigour ie. all the i's dotted and t's crossed, or some things left unsaid or which still remain to be proved.