Termination and exact output of the grid-complex algorithm

The validation stage scans the finite $n\times n$ marking array, counts the markings in each row and column, and traces a permutation on the finite set of $n$ markings. Each scan and each cycle check terminates. A failed size, marking, or disjointness condition returns invalid-grid; a valid marking array with more than one component cycle returns link-input-not-supported, exactly the two specified rejection branches.

On a valid knot input, the algorithm enumerates the $n!$ permutations. There are finitely many ordered state pairs; a pair has at most two candidate torus rectangles by [F2]. Testing whether a candidate is empty and counting its finitely many markings takes finitely many array operations. Every mode filter and coefficient calculation is therefore finite, so all matrix entries and grading lists are produced in finite time.

By [F2], every grid state appears exactly once in the permutation enumeration and every rectangle from a pair is among the enumerated torus rectangles. The empty-state test is the definition of ${\mathrm{Rect}}^{\circ }$; the mode-specific marking test and coefficient are the instructions in [F1]. Summing the accepted terms modulo two therefore gives exactly the specified differential matrix, and the remaining output fields are computed from the stated grading formulas.