The Computational Power of Typed Chronological Carry Refinement

The paper establishes the mathematical and computational power of typed chronological carry refinement (CCR-k), demonstrating how its intrinsic source chronology can be exploited to obtain provably optimal information representations, exact algorithms, and measurable computational advantages. Its central theorem determines the sharp minimum information required to answer every five-step future-source query, proving the exact archive law m(T,J,5) = 5J − 31 under explicit prime-field and accessibility conditions. Remarkably, this lower bound applies even to nonlinear deterministic encoders, and the paper constructs a causal archive attaining it exactly. It further derives a sparse future-response representation with at most 6J − 21 structurally admissible source monomials, characterizes finite-ring response profiles through exact lattice methods, and establishes optimal lossless encoding of realizable event-source masks, reducing their asymptotic representation cost from Θ(N²) to Θ(N log N) bits. Beyond these theoretical results, verified computational experiments demonstrate up to 8.62-fold faster multi-checkpoint forecasting against task-pruned Python dynamic programming, 12.7–14.5-fold faster response-profile counting, and up to 4.40-fold reductions in measured peak Python memory allocations in specified workloads. On real BPI2013 incident records, the lossless source representation reduces encoded payload size by 26.67%. Its main novelty is the combination of a sharp, nonlinear-encoder-resistant information lower bound, a matching constructive causal algorithm, exact arithmetic and graph-theoretic representations, and experimentally verified computational improvements. The results establish CCR-k as an algorithmically useful mathematical framework with provable information efficiency and task-dependent speed and memory advantages, without claiming universal superiority over all optimized computational methods.