Proof that the Universe is not a Turing Machine:

The undecidability of the Spectral Gap: Toby Cubitt (an appropriate name if there was ever one! He's almost a qubit!!), David Pérez-García, and Michael M. Wolf

https://www.nature.com/articles/nature16059

https://arxiv.org/abs/1502.04573

I think that this is one of the most important papers of the 21st century and all time.

Essentially Godel (Cantor, Zemleko, Peano, Russell, Turing) following the Hilbert Program (which followed Charles Babbage/Ada Lovelace and the Analytic and Differential Machines - should be able to crank the handle and get a new math proof, just as one can do for log or sine etc. tables) was/were saying that 1st order formal systems are limited. Obviously we can contemplate (to some extent, we are only human**, after all) infinity - such as a proof by induction. So we are definitely not 1st order/Turing Machines, NOR IS THE UNIVERSE.

Hint: it involves (as all of this stuff that deals with infinity, recursion, self-reference and the limits of 1st order formal systems) calculating the sequence of something like +1 -1 +1 -1 ... which may be some lattice potential.

** The first three Aleph transfinite numbers: Countably infinite, Uncountably infinite, Power Set of Reals (for example ALL the functions that can map a real number to another real number), I can "contemplate". This tower of infinities goes on... infinitely.