Autonomous tool-using agents in networked environments must decide which information source to query and when to stop querying and act. Without principled bounds on information-acquisition costs, unconstrained agents exhibit systematic failure modes: excessive tool use under congestion, prolonged deliberation under time decay, and brittle behavior under ambiguous evidence. We propose the Triadic Cognitive Architecture (TCA), a decision-theoretic framework that formalizes these failure modes via cognitive friction. By combining nonlinear filtering, congestion-dependent cost dynamics, and HJB optimal stopping, TCA models deliberation as stochastic control over a joint belief-congestion state, explicitly pricing information by tool signal quality and live network load. TCA yields an HJB-inspired stopping boundary and a computable rollout-based approximation of belief-dependent value-of-information with a net-utility halting condition. We validate TCA in two controlled environments (EMDG and NSTG) designed to isolate stopping quality, action selection under congestion, and temporal urgency. TCA improves resource outcomes while reducing time-to-action without degrading accuracy, gaining 36 viability points in EMDG and 33 integrity points in NSTG over greedy baselines. Ablations show that selection and stopping must be optimized jointly, as stopping rules alone recover at most 4 viability points. Sensitivity sweeps over alpha, beta, and lambda_S yield stable accuracy and interpretable trade-offs, and a continuation-value sweep over eta values 0, 0.1, 0.3, and 0.5 finds eta equal to zero is optimal under high temporal urgency. Finally, we demonstrate an illustrative instantiation around a black-box LLM on a memorisation-free corpus, where the same stopping principle executes using empirically computable uncertainty and value-of-information proxies.