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At the edge of stability, the iterates of gradient descent oscillate along the sharp directions of the Hessian. This poses a significant challenge for continuous-time models as the usual continuous-time approximation -- the gradient flow -- admits no oscillations. Several continuous-time models have been proposed to capture the behavior at the edge of stability, among them central flow and rod flow. Although both models can be interpreted as performing a coarse-graining in time, the connection between them remains poorly understood. Here we show that both flows can be derived from a third object, the edge coupling: a functional on consecutive iterates whose critical points are the fixed points and period-two orbits of gradient descent. It is revealed that the two flows correspond to coarse-grainings at different time scales. Rod flow coarse-grains over only the fastest time scale, while central flow coarse-grains over both the fast and intermediate time scales. Furthermore, using the edge coupling, we derive rod flows and central flows for momentum methods such as heavy ball and Adam. We verify the derived flows experimentally on representative neural network architectures, where they track the discrete iterates throughout all phases of training -- including the transition from the stable regime to the edge of stability.

Frontier language models are trained to exhibit stable programming assistant personas, yet those personas may drift over long agentic coding sessions in production. After a long real-world coding session, a model that hedges preferences in a fresh chat ("I don't have preferences") may instead express them directly ("Python -- the feedback loop is instant..."). Existing persona-stability studies focus on scripted chat of tens of turns, leaving long production coding sessions involving thousands of tool-using agent steps, repeated compaction, and hours of interaction largely uncharacterized. We introduce ContextEcho, a benchmark and reusable harness with a 25-probe identity suite, a snapshot-then-probe protocol that evaluates conversation state without perturbing the main session, judge-based and judge-free metrics, and a corpus of 57 long agentic-coding sessions collected with consent and redacted for release across Claude Code and Codex CLI. We use one representative session for a 23-model cross-provider study and all 57 sessions for corpus-scale trajectory analysis across frontier and open-weight models. Results show that persona drift occurs across providers and emerges early in real agentic interaction, while in-session compaction neither reliably resets nor amplifies it. A single-shot anchor restores the original assistant persona, though placebo controls suggest a generic register-reset effect. On same-task continuations, drifted prefixes improve tool-call fidelity -- which we attribute to retained task context rather than drift itself. In tool-free chat, however, the same prefixes break output-format contracts and inflate output length. These findings motivate ContextEcho as an open-source framework for auditing persona drift in long coding sessions without retraining.