User blog comment:B1mb0w/Fundamental Sequences/@comment-5529393-20160124015439/@comment-27173506-20160206193404

"it seems to imply phi(2,0) = phi(1,phi(2,0))". If that what it seems to imply, great! That's what it's supposed to imply. phi(2,0) is literally defined as the first alpha such that phi(1,alpha)=alpha. phi(2,a) is the 1+ath ordinal such that phi(1,b)=b. Your fundamental sequences imply that e_{{zeta_0}+1}=zeta_0+1, which is trivially false. Or you could define a new function, ignoring the whole fixed points issue, but that's not the Veblen function (and it's much weaker. phi(1,0,0) in your system is equal to phi(3,0) in the regular system).