Operator-Based Limits and Dynamics of Self-Attention as Contextual Search in Curved Value Space

Self-attention is conventionally described as using key--query similarity to weight and combine value vectors associated with context tokens. We develop an operator-based formulation in which attention performs contextual search over value-space directions. The factorisation identifies keys as activation functionals, values as response directions, and queries as their excitations. We derive a hierarchy between potential value space, contextual support, and responses reachable through key--query geometry. We show that responses within the span of the context's value vectors can be unreachable by attention. In causal decoders, each response reconfigures the accessible contextual region within value space. Prompting, continuation, and test-time thinking can alter access by reactivating associations, expanding support, or changing causal order. Successive head outputs form a value-space trajectory, while visible keys and values induce a response metric. This is related to the effective number of memory directions through the von Neumann entropy. We derive closed forms for the trajectory's displacements and discrete curvature. We find that the context-dependent metric curves output value space through an effective Riemann tensor. These distinctions predict causal signatures for missing representation, missing contextual support and failed access, providing a testable framework with new observables for semantic interpretability.