Temporal Sheaf Semantics for Inquiry and Agency: Genus-Regularized Procedures with Coherent Base Change

This paper extends the Topos-Semantic foundation of Paper I from static context to time-evolving behavior. Where Paper I models context as slice localization and compression as sheafification inside a fixed relevance fiber, Paper II temporalizes the entire noetic pipeline by indexing agent state, evidence, and admissibility by time. Agent behavior is treated as a sheaf-like object over a temporal site, so “inquiry” becomes the evolution of a section under restriction/transport, judgment, and commitment across time rather than merely the accumulation of tokens. A key contribution is the treatment of procedures (plans, experiments, tool queries, composite interventions) as diagrammatic objects whose execution is functorial transport through time-indexed relevance slices equipped with local modalities expressing admissibility/stabilization at each temporal focus. To control the global coupling created by procedural composition, the paper introduces a genus-like procedural entanglement invariant (with computable surrogates such as cycle rank, and optional ribbon-graph enhancements). This invariant serves as a regularizer that penalizes nonlocal dependency cycles that cannot be eliminated by local stabilization, yielding disciplined learning dynamics when balanced against conservative accommodation costs. The central technical result is a coherent base-change (Beck–Chevalley-type) principle: under explicit compatibility hypotheses for the temporal modalities, the commitment reflector commutes with relevance transport. This ensures factorization invariance (the semantics of executing a procedure is independent of how its underlying transport diagram is refactored) supporting a robust, implementation-agnostic semantics for agentic pipelines. The paper concludes with worked temporal toy examples demonstrating a characteristic “sphere-to-torus transition,” where a minimal additional coupling forces a jump in global entanglement, triggers accommodation, and alters the space of admissible future procedures. This will be known as Paper II.