TAOCP 7.2.2.2 Exercise 520
The proposed solution does not answer the stated exercise.
Section 7.2.2.2: Satisfiability
Exercise 520. ▶ [24] Every instance of SAT corresponds in a natural way to an integer programming feasibility problem: To find, if possible, integers $x_1, \ldots, x_n$ that satisfy the linear inequalities $0 \le x_j \le 1$ for $1 \le j \le n$ and
$$l_1 + l_2 + \cdots + l_k \ge 1 \qquad \text{for each clause } C = (l_1 \vee l_2 \vee \cdots \vee l_k).$$
For example, the inequality that corresponds to the clause $(x_1 \vee \bar{x}_3 \vee \bar{x}_4 \vee x_7)$ is $x_1 + (1-x_3) + (1-x_4) + x_7 \ge 1$; i.e., $x_1 - x_3 - x_4 + x_7 \ge -1$.
Sophisticated "IP solvers" have been developed by numerous researchers for solving general systems of integer linear inequalities, based on techniques of "cutting planes" in high-dimensional geometry. Thus we can solve any satisfiability problem by using such general-purpose software, as an alternative to trying a SAT solver.
Study the performance of the best available IP solvers, with respect to the 100 sets of clauses in Table 6, and compare it to the performance of Algorithm C in Table 7.
Verified: no
Solve time: 12m01s
Correctness
The proposed solution does not answer the stated exercise. Exercise 7.2.2.2.489 asks for a recurrence for $T_n$, the number of signed involutions on $n$ elements. The proposed solution instead solves a completely unrelated graph-theoretic problem about queen placements on an $8\times8$ chessboard.
The solution therefore fails the primary requirement of addressing the exact question. It introduces notation for $Q_8$, boundary sets, and Turton's question, none of which appears in exercise 7.2.2.2.489.
Even if the queen graph argument were internally correct, it would not constitute a solution to the requested recurrence problem.
Gaps and Errors
The entire setup is unrelated to the exercise. This is a Critical error because the solution addresses a different problem.
The proposed solution never defines signed involutions, never counts them, and never derives a recurrence for $T_n$. This is a Critical error because the central mathematical task is completely omitted.
The claimed use of enumeration is also unsupported. Statements such as "The enumeration proves that every other $8$-element subset has at least $112$ boundary edges" and "The exhaustive orbit check gives the upper bound $|\partial S|\le184$" provide no enumeration details, no generated cases, and no independent proof of the bounds. These would be Justification gaps even if the problem had been the queen graph problem.
Several displayed numerical conclusions are asserted without derivation. For example, the values $112$, $184$, $45$, $56$, $0$, and $8$ are not obtained by explicit counting in the text. This violates the requirement that bounds and counts be derived rather than merely stated. These are Justification gaps.
The final statement about Turton's question depends on the unproved claim that the minimum of $|\partial_{\rm out}S|$ is $45$. Since the argument relies only on an unshown computation, it is not a self-contained proof. This is a Justification gap.
Summary
The proposed solution is entirely for a different TAOCP exercise. It neither defines nor counts signed involutions and gives no recurrence for $T_n$. The response therefore fails the exercise before any issue of rigor or completeness can be considered.
VERDICT: FAIL, the solution addresses an unrelated queen graph optimization problem instead of deriving a recurrence for signed involutions.