Complete Noncoprime Classification of Minimal Kernel Dimension for Rational Floor Maps

Paper XIII completes the noncoprime classification of minimal kernel dimension for rational floor maps in the fixed-control balance-loop model.

Let d = gm and q = gn, where g ≥ 2, 1 ≤ n < m, and m and n are coprime. The paper determines exactly when the partial floor map F_d,q(x) = floor(qx/d), with multiples of d deleted, admits an exact proper realization of kernel dimension 2, 3 or 4.

The complete classification is remarkably rigid. The exceptional map (d,q) = (4,2) has kernel dimension 2. Kernel dimension 3 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 reduced noncoprime case, 1 < n < m − 1, has kernel dimension 4. Thus the universal kernel-four upper bound is attained throughout the interior.

The principal new result is this interior maximality theorem. After bulk compression, any hypothetical kernel-three realization has four positive-control selectors. The proof classifies all such architectures according to the number of pure-control selectors. Two-pure architectures are reduced to two-generated numerical semigroups and excluded using Frobenius symmetry, Apéry coordinates, mechanical-word return structure and checkerboard substitutions. One-pure architectures are eliminated using residue rigidity, flat and rising edge structure, and bidirectional target shifts. Zero-pure architectures reduce to unimodular lattice paths and finite orthant obstructions.

The paper also gives an explicit five-column kernel-three realization for the previously unresolved co-affine boundary n = m − 1, together with computational verification of representative constructions and the finite lattice classifications used in the proof.

Conceptually, the invariant has a natural algebraic interpretation. For a fixed rank-two source/control configuration, kernel dimension equals the rank of its integer relation lattice, equivalently the codimension of the associated toric ideal. The problem can therefore be viewed as determining the minimum relation-lattice freedom, or restricted toric codimension, compatible with exact labeling of an entire fixed-control fibre by a rational floor map.

The result sits at the intersection of integer factorization, affine and numerical semigroups, toric geometry, Christoffel and mechanical words, and mixed-integer formulation complexity. General floor-map representability is easy in broader Presburger and mixed-integer models; the difficulty here comes from the much stronger requirement that every nonnegative factorization on a fixed-control slice must produce the correct target.