Streaming incident diagnosis requires integrating long telemetry streams—logs, traces, alerts, metrics, tickets, and operator actions—under a context budget that is much smaller than the event history. Existing long-context mechanisms improve token capacity, but they do not directly specify which delayed failure dependencies should be preserved when the stream is noisy, drifting, and only partially observed. We present Causal-Dependency State Space Prompting (CSSP), a bounded-memory framework that treats diagnosis as online failure-dependency state estimation rather than raw prompt extension. CSSP combines (i) an event canonicalizer that normalizes templates, typed entities, and high-cardinality identifiers, (ii) a selective state-space memory trained with weak dependency supervision derived from request identifiers, entity co-reference, and template-transition rules, and (iii) a learned state-token interface that exposes the latent state to the same frozen 7B instruction-following reasoner used by all large language model baselines. We explicitly soften the causal claim: the supervision does not prove interventional causality, but encourages the memory to preserve operational dependency evidence relevant to diagnosis. A constrained verifier checks evidence alignment, tool consistency, and unsafe actions with one bounded revision step. Across BGL, HDFS, and OpenStack, CSSP obtains the best mean 1 with five-run confidence intervals, remains robust to log corruption and noisy weak links, and explains the smaller OpenStack gain through incident-type analysis. These results support dependency-aware state tokenization as a practical interface for reliable bounded-memory incident reasoning.