Exact Future-Archive Complexity and Exceptional-Characteristic Recovery in Typed Chronological Carry Refinement

The paper establishes a complete mathematical theory of optimal future-information preservation in typed chronological carry refinement (CCR-k), determining exactly how much historical information must be retained to reconstruct every possible future response under arbitrary source programs. Its central theorem gives the sharp minimum archive complexity for every admissible observation horizon, expressing it as an exact direct-read information count plus the rank of an explicit five-column transfer matrix. Crucially, this minimum is proved against all deterministic encoders, including nonlinear ones, and is attained by a constructive causal archive. Beyond extending the earlier five-step CCR result to arbitrary finite horizons, the paper provides a complete classification of exceptional finite-field characteristics, identifying precise rank deficiencies at horizons four, five, and six, including exceptional prime–checkpoint pairs and an exact degree-six polynomial criterion. It further proves a striking recovery phenomenon: every exceptional boundary-rank deficiency disappears by horizon seven. An exact saturation theorem then establishes the ultimate archive complexity \(5+J(J-7)/2\), showing that sufficiently long admissible observation horizons require no additional historical information. Its main novelty is the exact characterization of the interplay between source chronology, finite-field arithmetic, future observability, and irreducible information complexity, combining sharp information-theoretic lower bounds, matching causal constructions, complete exceptional-characteristic classifications, and a finite-horizon recovery theorem. The results extend CCR-k from a fixed-horizon compression theorem to an exact all-horizon theory of future-response equivalence and optimal historical information preservation.