Proof of Putnam 2005 A4 (Appendix B) is likely unfixably wrong

[j-bagel]

Hi all,

Proof of Putnam 2005 A4 in Appendix B does not pass on my end, it maybe a different in terms of settings (maybe not), but I would like to point out that the proof is more than likely wrong and can't be fixed.
This problem has a very straightforward proof: notice that nI - S^T S is positive semi-definite, and we are done.
The proof in your paper tries to do some "Cardinal" arithmetic, I don't think could help with this (or these kind of) matrix problem.

(It is known that Lean4 REPL, in strange cases, could wrongly output pass. Since the argument that "we identified a distinctive pattern in its reasoning approach: the 7B model frequently employs Cardinal.toNat and Cardinal.natCast_inj to handle problems involving finite cardinalities" is quite important in achieving 62 out of 658 in the very difficult PutnamBench, is it possible to manually check similar proof to make it extra sure?)