The Paradox of Robustness: Decoupling Rule-Based Logic from Affective Noise in High-Stakes Decision-Making

While Large Language Models (LLMs) are widely documented to be sensitive to minor prompt perturbations and prone to sycophantic alignment, their robustness in consequential, rule-bound decision-making remains under-explored. We uncover a striking "Paradox of Robustness": despite their known lexical brittleness, aligned LLMs exhibit strong robustness to emotional framing effects in rule-bound institutional decision-making. Using a controlled perturbation framework across three high-stakes domains (healthcare, finance, and education), we find a negligible effect size (Cohen's h = 0.003) compared to the substantial biases observed in analogous human contexts (h in [0.3, 0.8]), approximately two orders of magnitude smaller. This invariance persists across eight models with diverse training paradigms, suggesting the mechanisms driving sycophancy and prompt sensitivity do not translate to failures in logical constraint satisfaction. While LLMs may be "brittle" to how a query is formatted, they appear considerably more stable against affective attempts to bias rule-bound decisions. To probe the boundary of this finding, we add two reviewer-driven side studies. A five-scenario immigration extension yields a small but statistically detectable +0.8 percentage point shift that remains within a pre-specified +/-3 percentage point Region of Practical Equivalence (ROPE), while a screening-level adversarial narrative pilot finds no meaningful decision shift under stronger LLM-generated prompts. We release a core benchmark (9 base scenarios x 18 condition variants = 162 unique prompts), code, and data to facilitate replicable evaluation.