Can you explain more formally what you mean by "proves that it itself exists"?

The fundamental principle of Syntacticism is that the derivations of a formal system are fully determined by the axioms and inference rules of that formal system. By proving that the ordinal kappa is a coherent concept, I prove that PA+kappa is too; thus the derivations of PA+kappa are fully determined and exist-in-Tegmark-space.

Actually it's not PA+kappa that's 'reflectively consistent'; it's an AI which uses PA+kappa as the basis of its trust in mathematics that's reflectively consistent, for no matter how many times it rewrites itself, nor how deeply iterated the metasyntax it uses to do the maths by which it decides how to rewrite itself, it retains just as much trust in the validity of mathematics as it did when it started. Attempting to achieve this more directly, by PA+self, runs into Löb's theorem.