Cascade-free sequences, dispersion index, and state avoidance for stateful digit-wise operations

Abstract

We show that cascade-free counting from carry theory is a special case of a general transfer matrix construction. For any binary stateful digit-wise operation with GEN/PROP/KILL decomposition, the number of cascade-free sequences of length $L$ depends on only two parameters: the alphabet size $N$ and the product $d = |\text{GEN}| \cdot |\text{PROP}|$. The resulting sequence satisfies $a(L) = N a(L-1) - d a(L-2)$ and equals a scaled Chebyshev polynomial of the second kind with coupling parameter $x = N/(2\sqrt{d}) \geq 1$. We instantiate this for digit-wise addition and doubling in base $p$. For odd primes the exact relation $a_{\text{carry}}(L) = p^L a_{\text{dbl}}(L)$ holds. For $p = 3$ the cascade-free doubling count equals the Fibonacci bisection $F(2L+2)$ via $U_L(3/2) = F(2L+2)$ (OEIS A001906); we are not aware of this interpretation in the existing literature. We analyse the dispersion index $D = \text{Var}(\nu)/E[\nu]$ of the state count for uniformly distributed inputs. For symmetric chains ($g = k$) the Poisson transition $D_\infty = 1$ occurs at $\mu = 1/3$, corresponding to base 3 where the Fibonacci bisection appears. The finite Poisson transition point $\mu^*(L)$ decreases strictly to $1/3$ with rate $1/(6L) + O(1/L^2)$. We generalise to state spaces $|S| > 2$ via state avoidance. The restricted transfer matrix has dimension $s-1$; the Chebyshev representation persists for $|S| = 3$.