System prompts for AI coding agents increasingly employ motivational framing -- from neutral task descriptions to fear-driven threats -- yet no controlled study has examined whether such framing affects agent behavior. We present two studies investigating how trust-based versus fear-based motivation framing in system prompts influences AI agent debugging performance. In Study 1, we conducted a controlled manual experiment comparing a trust-framed methodology (NoPUA) against an unframed baseline across 9 debugging scenarios using Claude Sonnet 4. Trust-framed agents found 59% more hidden issues (p = 0.002, d = 2.28) while taking 83% more investigative steps, despite finding 15% fewer surface-level issues -- revealing a depth-over-breadth tradeoff in investigation strategy. In Study 2, we replicated and extended these findings with 5 independent automated runs across 3 conditions (Baseline, NoPUA trust-framed, PUA fear-framed), yielding 135 scenario-level data points. Trust-framed agents again showed significant advantages: +74% investigative steps (p = 0.008) and +25% hidden issues found (p = 0.016). Crucially, fear-framed (PUA) agents showed no significant improvement over baseline on any metric (all p > 0.3), demonstrating that fear-based motivation is ineffective for AI agents. We ground these findings in Self-Determination Theory, regulatory focus theory, and satisficing models, arguing that trust-based framing induces exploration-oriented, promotion-focused behavior while fear-based framing fails to shift agents from default satisficing strategies. Our results suggest that the motivational frame of system prompts -- not just their technical content -- causally influences AI agent investigation depth.