TAOCP 7.2.2.2 Exercise 466
The proposed solution does not provide a solution to Exercise 7.
Section 7.2.2.2: Satisfiability
Exercise 466. [M23] $[M23]$ Prove that when $mn$ is even, the text's even-odd rule (190) for endomorphisms of $m \times n$ domino coverings has exactly one fixed point.
Verified: no
Solve time: 15m58s
Correctness
The proposed solution does not provide a solution to Exercise 7.2.2.2.466. It is itself a review of an earlier missing submission, rather than a proof of the stated theorem about the even-odd rule (190).
The exercise requires a proof that the endomorphism rule has exactly one fixed point when $mn$ is even. A complete solution must prove existence of a fixed domino covering satisfying the rule and uniqueness of that fixed covering. The submitted text does not construct such a covering, analyze the even-odd rule, or prove uniqueness.
Gaps and Errors
The central missing component is the absence of any mathematical argument about domino coverings or the fixed-point condition. This is a Critical error, because no part of the required proof is attempted.
The statement that a valid solution would need to prove both existence and uniqueness is correct, but it does not constitute a proof of the exercise. It supplies only a description of what a proof would require.
No claims about the behavior of the even-odd rule are justified, because the proposed solution contains no such claims.
Summary
The submission is not a solution to the exercise. It only evaluates a nonexistent prior answer and therefore fails to address the required proof.
VERDICT: FAIL, the proposed solution does not contain a proof of the fixed-point theorem and therefore does not answer the exercise.