\begin{abstract}
The agent harness---the system layer comprising prompts, tools, memory, and orchestration logic that surrounds the model---has emerged as the central engineering abstraction for LLM-based agents. Yet harness design remains ad~hoc, with no formal theory governing composition, preservation of properties under compilation, or systematic comparison across frameworks. We show that the categorical Architecture triple $(G, \mathrm{Know}, \Phi)$ from the ArchAgents framework provides exactly this formalization. The four pillars of agent externalization (Memory, Skills, Protocols, Harness Engineering) map onto the triple's components: Memory as coalgebraic state, Skills as operad-composed objects, Protocols as syntactic wiring~$G$, and the full Harness as the Architecture itself. Structural guarantees---integrity gates, quality-based escalation, supported convergence checks---are $\mathrm{Know}$-level certificates whose preservation is structural replay: our compiler checks identity and verifier replay, not output-layer correctness or model behavior. We validate this correspondence with a reference implementation featuring compiler functors targeting Swarms, DeerFlow, Ralph, Scion, and LangGraph: the four configuration compilers preserve three named certificate types by identity/replay, and LangGraph preserves the same certificates through its shared per-stage execution path. The LangGraph compiler creates one node per stage using the same per-stage method as the native runtime, providing LangGraph-native observability without reimplementing harness logic. An end-to-end escalation experiment with real LLM agents confirms that the quality-based escalation control path is model-parametric in this two-model, one-task experiment. The result positions categorical architecture as the formal theory behind harness engineering.
\end{abstract}
