Residual Streams in Gradient Flows: Drift-Slew Decomposition of Optimization Residuals as a Convergence Diagnostic and Saddle-Proximity Indicator

We establish that the iteration residual stream of gradient descent, when decomposed through the Drift-Slew Fusion Bootstrap (DSFB) framework, encodes the spectral structure of the objective's Hessian and provides a zero-cost convergence diagnostic and saddle-proximity indicator. This paper is the third in the DSFB residual primacy series: Paper I (DOI: 10. 5281/zenodo. 19382012) established exact residual spectral identities for linear Laplacian relaxation; Paper II (DOI: 10. 5281/zenodo. 19410239) extended the framework to nonlinear non-convergent Hamiltonian dynamics. The present paper treats the nonlinear convergent case and closes the series with a synthesis theorem stating residual primacy across the dynamical spectrum. The central exact result is: for gradient descent on a strongly convex quadratic with step size h < 2/λₘax, the DSFB drift-slew decomposition of the iteration residual is exact, with each eigenmode of the Hessian contributing a geometrically decaying component. The ratio of drift to slew decay rates is an exact closed-form function of the extreme Hessian eigenvalues, recovering the condition number κ (A) exactly without computing any eigenvalues. This positions DSFB as a streaming, causal spectral method for symmetric positive definite matrices: condition number recovery at O (n) cost per step, without orthogonalization or storage of past iterates. For general smooth strongly convex objectives, the Hessian varies along the trajectory, making the spectral identity approximate and asymptotically exact as convergence is reached. The bootstrap residual β (t) is proposed as a saddle-proximity indicator: provably O (h) at strongly convex minima, and conjectured to grow near non-degenerate saddle points where mixed-sign Hessian eigenvalues drive the iteration matrix outside the unit disc. No numerical experiments are reported and no empirical claims are made. All claims beyond the quadratic spectral identity and the zero-baseline corollary are presented as conjectures or directions for future experimental investigation. Keywords: gradient descent; gradient flow; DSFB; drift-slew decomposition; optimization residual;condition number; saddle point; convergence diagnostic; residual primacy; endoduction;Riemannian gradient flow; conjugate gradient