Rank-Wise Approximations of Hyper-Operations

[Daniel]

Yes, that is perfectly possible.

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This implicit solution exists uniquely across iterations. But if you ask for a tetration solution, it's not enough to declare uniqueness. Even while moving your base value, it's not enough.

There are infinite solutions to these equations. This formula converges as your describing. But there's little to no general uniqueness. There are countably infinite solutions, and just because an algorithm evaluates to something, that doesn't qualify as a uniqueness condition.

JmsNxn
Yes, I agree with you, except I believe there are at least \( \aleph_1 \) solutions, the countable infinity of period 1 fixed points, the uncountable infinity of n-period fixed points like the period 2 fixed point of \( 0^0=1, 0^1=0 \). Then there are the countable infinite rationally neutral fixed points on the Shell Thron boundary.