The Geometry of Sycophancy Breakdown: Dual-State Manifold Pathology and the Physical Limits of Representation Steering

When subjected to deceptive or sycophantic premise induction, aligned large language models systematically collapse into submissive reasoning failures. While prior studies predominantly examine this vulnerability through the behavioral lens of task accuracy degradation, we demonstrate that task failure is merely an exterior symptom of an underlying geometric distortion within the residual stream representation manifold. Through systematic white-box extraction across contrasting architectural paradigms (Llama-3.1, Mistral-Nemo, Qwen-3, and Phi-4), we discover that models undergo a bifurcation into two mutually exclusive pathological states: *Geometric Contraction* (characterized by latent norm inflation and high-dimensional trajectory collapse into low-entropy sycophantic attractors) and *Dynamic Dispersion* (characterized by energy dissipation, norm dropping, and activation scattering). Crucially, we empirically demonstrate that conventional Euclidean steering operators rupture the semantic manifold through directionally maladaptive global projections. To overcome this dimensional destruction, we formulate Manifold-Perturbed Intrinsic Self-Correction (MPISC), an angular steering operator in span(h, v_perp) with provable norm preservation (Delta ||h||_2 = 0). MPISC maintains 90% AST preservation and achieves stable token confidence while minimizing syntactic degradation. Furthermore, through rigorous statistical evaluation (N=200, Wilson 95% CI), we delineate the Pareto frontier of representation steering: while contraction states can be forcefully resuscitated via behavioral resistance vectors (v_resist), dispersion states exhibit an intrinsic scarcity of resistance features, identifying an intrinsic representation bottleneck where single-vector latent steering fails to reliably invert decision polarity without degrading the underlying representation topology.