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…