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

Paper XII. Minimal Kernel Dimension for Fixed-Control Balance-Loop Realizations of Rational Floor Maps. Preprint, August 2026.

This paper introduces the minimal kernel dimension of an exact fixed-control balance-loop realization of a rational floor map. For the partial map x ↦ floor(Qx/d), defined on integers not divisible by d, the invariant κ(d,Q) measures the least relation-lattice dimension, equivalently the least toric codimension, required by any proper realization.

The paper first proves a rigidity theory for minimal presentations. Every zero-control column is forced to lie in the bulk direction (d,Q), and every kernel-minimal realization contains exactly one bulk column. This reduces the infinite arithmetic realization problem to a finite selector-fibre problem.

For coprime parameters the invariant is classified completely. Kernel dimension 2 occurs exactly when Q is congruent to +1 or −1 modulo d; every other coprime case has kernel dimension 3.

The noncoprime case exhibits genuinely new behaviour. In particular, κ(8,2) = 3 while κ(12,3) = 4, even though both maps have the same reduced slope 1/4. The scaling family (d,q) = (mg,g), with m ≥ 3 and g ≥ 2, is classified completely: the kernel dimension is 3 when m = 3 or g = 2, and 4 when m ≥ 4 and g ≥ 3.

The paper then proves a general kernel-4 realization theorem for arbitrary noncoprime parameters d = gm and q = gn with gcd(m,n) = 1. Combined with the coprime classification and the scaling constructions, this yields the sharp universal bound

κ(d,Q) ≤ 4

for every d ≥ 2 and every positive numerator Q. Since κ(12,3) = 4, the universal bound is best possible.

The lower bounds combine lattice saturation, signed-residual arguments, residue replacement, numerical semigroups, Apéry coordinates and rational mechanical words. The upper bounds are supplied by explicit fixed-control constructions. The results show that minimal factorization complexity is universally bounded, but is not determined by reduced rational slope alone; it also depends on the arithmetic structure of the excluded congruence class.

The paper also identifies a toric interpretation of the invariant: κ(d,Q) is the minimum height of the toric ideal associated with the source matrix, taken over all exact fixed-control realizations of the prescribed floor map.