Abstract
The companion paper established that carry depth under incrementation in canonical positive linear
recurrence numeration systems (PLRS) converges to a geometric distribution determined by the
Perron root of the defining recurrence. The present paper studies the arithmetic cost of those
carries.
A new structural theorem is proved showing that the lexicographically maximal legal words form
the prefixes of a single infinite periodic word
P∞,
P =c1c2···cL−1(cL −1),
for every canonical coefficient vector c = (c1,…,cL). Combined with the canonical successor
decomposition established in the companion paper, this shows that every additive carry statistic is
obtained by evaluating an explicit periodic boundary map on the carry depth.
A general pushforward theorem is established for periodic boundary maps, reducing the limiting
distribution of every additive carry statistic to the geometric carry-depth law. Two natural statistics
are then studied in detail: Hamming carry cost, which counts the number of digit positions changed
by incrementation, and digit-magnitude (ℓ1) carry cost, which records the total digit mass erased
during the carry.
Exact limiting distributions, rational generating functions, and reconstruction procedures are
obtained for both statistics. The Hamming carry law determines the complete support sequence
1
of the maximal periodic boundary word, while the digit-magnitude law determines the complete
infinite boundary word itself. Whether this always determines the canonical reduction of the defining
recurrence in the sense of the companion paper is identified as an open problem. A simple identity
derived from the characteristic equation yields the universal expectation
E[Cℓ1] = 2
for every canonical PLRS, extending the classical mean-two phenomenon beyond the binary setting.
Together with the companion paper, these results separate the probabilistic geometry of carry
propagation from its deterministic arithmetic boundary structure.
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