Minimal Kernel Dimension for Fixed-Control Balance-Loop Realizations of Rational Floor Maps

This paper introduces the minimal kernel dimension of an exact fixed-control balance-loop realization of a rational floor map. For the partial map that sends x to the floor of Qx/d on integers not divisible by d, the invariant measures the least relation-lattice dimension, equivalently the least toric codimension, required by any proper realization. The paper first proves structural rigidity results for minimal presentations. Every zero-control column must be a positive multiple of the bulk direction, and every kernel-minimal realization can be reduced to a single bulk column together with a finite selector fibre. For coprime parameters, the invariant is completely classified: kernel dimension 2 occurs exactly when the numerator is congruent to plus or minus 1 modulo d, while all other coprime cases have kernel dimension 3. The noncoprime case shows genuinely different behaviour. In particular, the paper proves that the maps with parameters (8,2) and (12,3) have kernel dimensions 3 and 4 respectively, even though both have reduced slope 1/4. More generally, the complete scaling family with parameters (mg,g), for m at least 3 and g at least 2, is classified: the kernel dimension is 3 when m equals 3 or g equals 2, and 4 otherwise. The proofs combine lattice saturation, residue arguments, numerical semigroups, Apéry coordinates, rational mechanical words, and explicit fixed-control constructions. The results show that minimal factorization complexity depends not only on the reduced rational slope, but also on the arithmetic structure of the excluded congruence class.