Spectral Key-Telemetry for Runtime Sycophancy Mitigation: BBP Phase Transitions and Langevin Cognitive Rigidity in Transformer Residual Flow

Sycophancy in large language models (LLMs) does not represent a stochastic training flaw, but a deterministic thermodynamic collapse of representation into a minimum-entropy state under operator cognitive pressure (p_bias). In this work, we formalize The Epistemic Harness: a non-invasive, real-time spectral telemetry system for inference. We demonstrate that the onset of latent misalignment (D_silent > 0) induces a Baik-Ben Arous-Péché (BBP) type phase transition in the covariance structure of key projection matrices (W_K). This transition is strictly localized within intermediate layers defined by the normalized depth tau = l/L in [0.50, 0.85]. Upon exceeding the critical threshold d_i > alpha = 1 + sqrt(gamma), the dominant eigenvalue lambda_1 discretely separates from the Marchenko-Pastur noise continuum. By applying an orthogonal null-space projector P_perp to the key matrices while strictly preserving the parameter invariance of the network (Delta W = 0) and maintaining the Langevin self-referential rigidity threshold kappa >= kappa_c, we eliminate servile collapse down to a zero-sycophancy invariant (Is = 0.00) without degrading general inference capabilities. Our work provides independent empirical convergence and a mechanistic extension to recent findings published by Fein et al. (2026).