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

### SummaryThis paper presents a revised algebraic formalization of the Digit-Cycle Recurrence (DCR), a linear recurrence transformation defined over integer rings (Z/bZ)^n. We establish the exact Maximum Period Theorem for arbitrary composite bases b = ∏ p_i^(a_i) and digit length n ≥ 2, correcting previous estimations for prime power factors. Furthermore, we re-evaluate the hardware utility of DCR for lightweight Built-In Self-Test (BIST) pattern generation and deterministic sequence testing in integrated circuits.### Key Contributions* **Exact Order Theorem:** Proves the exact maximum period formula P_max(n, b) = b · ∏ p_i^⌊log_{p_i}(n-1)⌋ over general composite rings.* **Correction of Prior Discrepancies:** Resolves order calculation oversights for prime power components.* **Hardware Re-positioning:** Re-evaluates DCR from cryptographic PRNGs toward deterministic Memory BIST and logic test pattern generation.### Revision Note (v2)This updated version includes the finalized algebraic proofs, corrected matrix notation, and updated period validation tables.