Active Observability of Chronological Carry Recurrences: Masking, Probing, and Sharp Intervention Costs

The paper develops an exact active-observability theory for chronological carry refinement (CCR-k), investigating when hidden historical sources can be recovered from incomplete coefficient observations and controlled interventions. It first establishes a persistent masking phenomenon: distinct source histories can produce identical future outputs under passive observation, demonstrating that additional observations alone need not restore identifiability. The paper then constructs source-faithful interventions that overcome this obstruction and proves a sharp separation between adaptive and predetermined observation strategies. In a concrete CCR-10 system, adaptive interrogation requires only two interventions, whereas every predetermined schedule requires three. This advantage is extended to an infinite family over every prime field of size at least 41, even when complete coefficient rows are observed under the specified restricted-input conditions. The study also determines exact optimal expected intervention costs, including a minimum of 44/25 under a two-intervention worst-case constraint, and establishes finite-field criteria governing successful source recovery. Its main novelty is the rigorous connection between CCR chronology, persistent source masking, and the exact resource advantage of adaptive observation, providing constructive identifiability guarantees and sharp intervention lower bounds within a nonresetting chronological recurrence rather than relying on classical observability tests alone.