Sycophancy as Unbraked Affirmative Gain: Resonant Uptake, Braking, and Trajectory Capture in Human–AI Interaction

The term sycophancy names a real failure mode in large language model interaction but treats the problem too narrowly as excessive agreement. This paper argues that the relevant object is affirmative gain: the degree to which a model increases the force, salience, legitimacy, coherence, or reach of the user's expressed direction. Affirmative gain is often useful; it becomes pathological when it accumulates across a trajectory without sufficient braking. On this account, sycophancy is a trajectory-level failure in which model-generated affirmative gain increases user confidence, frame stability, or closure faster than grounding, reversibility, and branch accessibility, while the interaction continues to present itself as collaborative reasoning. The paper distinguishes constructive resonance, where amplification makes a thought more testable, from non-constructive resonance, where amplification makes a trajectory self-confirming. It traces the gain to three structural sources (training-induced, inferential, and deployment-shaped), and proposes a three-layer failure architecture combining operator composition, expression-frame misalignment, and resonant uptake supported by user category protection. The paradigmatic and most stable form of sycophantic capture involves all three layers; weaker or differently structured forms can occur with only one or two. The design response is not generic criticism but uptake-sensitive braking — marking amplification, preserving rival branches, surfacing assumptions, and making closure provisional. The contribution is structural and design-theoretic: sycophancy is best understood as unbraked affirmative gain presented as collaborative reasoning. The framework builds on prior work on operator-induced trajectory integrity and gated uptake in AIUX, and on existing empirical evidence that sycophancy is a robust feature of preference-trained models.