Digit-Cycle Recurrence (DCR): Maximum Period, Ring-Theoretic Properties, and Deterministic Hardware Primitives

AbstractThis paper presents an algebraic formulation of the Digit-Cycle Recurrence (DCR), a linear recurrence transformation over the finite ring Z_b=Z/bZ. For a state length n≥2 and an arbitrary composite base b=∏_{i=1}^{r}p_i^{a_i}, we derive the exact multiplicative order of the associated lower-unitriangular prefix-sum matrix. For each prime-power component p^a, the order is p^{a+floor(log_p(n-1))}; consequently, the maximum period over Z_b^n is P_max(n,b) = b · ∏_{i=1}^{r}p_i^{floor(log_{p_i}(n-1))}. The result gives an exact algebraic period bound for arbitrary composite bases. We also describe the resulting cycle structure and discuss DCR as a deterministic, low-complexity primitive for hardware test-pattern generation and Built-In Self-Test (BIST), where predictable state transitions may be preferable to pseudorandom stimulation.KeywordsDigit-Cycle Recurrence; finite rings; unitriangular matrices; multiplicative order; prime-power modulus; cycle structure; BIST; deterministic test-pattern generation.Revision note. This consolidated version corrects the exponent in the prime-power order theorem and propagates the correction through the composite-base formula and validation examples. The correction changes the logarithmic exponent from a ceiling to a floor.Author note. This version is a corrected and consolidated mathematical presentation of the DCR manuscript. The principal correction is the use of floor (log_p(n-1)) in the exact order exponent.