Residual Trajectories as Primary Epistemic Objects: Drift-Slew Decomposition of Laplacian Relaxation and the Spectral Fingerprint of Graph Structure

We propose that the residual trajectory of Tutte's spring embedding, when analyzed through the drift-slew decomposition of the Drift-Slew Fusion Bootstrap (DSFB) framework, carries structural spectral information about the graph Laplacian that is discarded by classical convergence-oriented treatments - and argue that this trajectory, not the fixed-point embedding, is the epistemically primary object of the problem. This paper is a conceptual and theoretical contribution: no numerical experiments are reported and no empirical claims are made. The core argument proceeds from a spectral identity established in Section 3, relating the residual sequence to the Laplacian eigendecomposition, which is mathematically exact under the stated assumptions. All downstream claims regarding spectral fingerprinting, graph similarity, and safety-critical applicability are presented as conjectures or directions for future experimental investigation, not as established results. We further propose that this reframing instantiates a mode of reasoning we term endoduction in which internal residual organization, rather than external boundary conditions or model hypotheses, is the primary source of structural inference. The paper's purpose is to establish the conceptual framework and stake the theoretical claims precisely; experimental validation is deferred to subsequent work. Tutte embedding; Laplacian relaxation; spectral graph theory; drift-slew decomposition; DSFB; residual trajectory; endoduction; graph fingerprinting