Boundary Maps and Travelling Carry Layers in Binary Positive Linear Recurrence Numeration

Abstract

We study Hamming carry propagation under incrementation in a one-parameter family of binary positive linear recurrence numeration systems. For integers k ≥ 2 and 1 ≤ a < k, the defining coefficient vector is

c_k,a = (1, …, 1, 0, …, 0, 1),

with k − a initial 1s and a − 1 zeros. This family interpolates between the canonical k-bonacci recurrence and systems whose maximal legal representations contain longer zero blocks.

The principal object is the boundary carry map, which assigns to each maximal legal suffix the number of digits changed when that suffix is reset during incrementation. We show that this map possesses exceptional fibres of cardinality a + 1, producing isolated spikes in the limiting carry distribution. The limiting law is obtained as the pushforward of an explicit geometric distribution under the boundary map:

p_k,a(r) = (1 − q) Σ_Dk,a(m)=r q^m,

where q ∈ (0,1) is the reciprocal of the dominant characteristic root.

Despite the presence of boundary collisions, the limiting mean Hamming carry length satisfies the exact identity

μ_k,a = 2

for every member of the family. An exact closed-form variance formula is likewise proved for every admissible pair (k, a), by two independent derivations in Section 10 and Appendix B. Its fixed-a asymptotic expansion as k → ∞ is developed and proved in Section 11, including rigorous error control at every fixed boundary coordinate.

The travelling boundary layer found previously for canonical k-bonacci numeration is shown to arise from the non-injectivity of the boundary carry map. The parameter a provides explicit control over both the location and the amplitude of this phenomenon.

The results identify boundary-map geometry, rather than the recurrence relation alone, as the mechanism governing higher-order carry statistics. The structural theory, exact finite carry recurrence, exact mean and exact variance are proved in full for every admissible pair (k, a), together with the fixed-a asymptotic consequences of these formulas.