Complete Noncoprime Classification of Minimal Kernel Dimension for Rational Floor Maps

This paper completes the classification of minimal kernel dimension for genuinely noncoprime rational floor maps in the fixed-control balance-loop model. For reduced noncoprime data d = gm and q = gn, with g at least 2, 1 <= n < m, and m and n coprime, the paper determines exactly when the minimum kernel dimension is 2, 3 or 4. There is a single kernel-two exception, the map with d = 4 and q = 2. Kernel dimension three occurs on the co-affine boundary n = m - 1, apart from that exception, and on the scaling boundary n = 1 when either g = 2 or m = 3. Every genuinely interior case, with 1 < n < m - 1, has kernel dimension four. The main new result is therefore an interior maximality theorem: every reduced noncoprime interior staircase requires the universal maximum kernel dimension four. The proof classifies all hypothetical kernel-three presentations according to the number of pure-control selectors. The two-pure case is reduced to a two-generated numerical semigroup and excluded using Frobenius symmetry, Apery coordinates, mechanical-word return structure and checkerboard substitutions. The one-pure cases are eliminated through residue rigidity, flat and rising edge structure, and bidirectional target-shift arguments. The zero-pure cases reduce to unimodular lattice paths and finite orthant obstructions. The paper also gives an explicit kernel-three construction for the previously unresolved co-affine boundary and provides computational verification of representative constructions and finite classification tables. Conceptually, the invariant can be interpreted as a restricted formulation-complexity measure. For a fixed rank-two source/control configuration, kernel dimension equals the rank of its integer relation lattice, equivalently the codimension of its associated toric ideal. The classification therefore determines the minimum relation-lattice freedom required for an exact fixed-control realization of a noncoprime rational floor graph.