Deploying AI agents for repetitive periodic tasks exposes a critical tension: Large Language Models (LLMs) offer unmatched flexibility in tool orchestration, yet their inherent stochasticity causes unpredictable failures, and repeated invocations incur prohibitive token costs. We present the LOOP SKILL ENGINE, a system that achieves a combined 99% success rate and 99% token reduction for periodic agent tasks through a one-shot recording, deterministic replay paradigm. On its first run, the agent executes the task with full LLM reasoning while the system transparently intercepts and records the complete tool-call trajectory. A greedy length-descending template extraction algorithm then converts this recording into a parameterized, branch-free Loop Skill -- a deterministic execution plan that captures the task's functional intent while parameterizing time-dependent and result-dependent variables. All subsequent executions bypass the LLM entirely: the engine resolves template variables against real-time values and replays the tool sequence deterministically. We prove two theorems: (1) Replay Determinism -- the step sequence of a validated Loop Skill is invariant across all future executions; (2) Write Safety -- concurrent access to persistent configuration is serialized through reentrant locks and atomic file replacement. Across a benchmark of periodic agent tasks spanning intervals from 5 minutes to 24 hours, the Loop Skill Engine reduces monthly token consumption by 93.3%--99.98% and cuts execution latency by 8.7x while eliminating output non-determinism. A multi-layer degradation strategy guarantees that tasks never stall. We release the engine as part of the buddyMe open-source agent framework.