Does Mandt, Hoffman & Blei (2017) actually model constant SGD as an Ornstein–Uhlenbeck process, and under what assumptions does 'SGD = OU near a minimum' hold?

Raised while promoting 10-inbox/raw/2026-07-11-hop-stasis-is-an-ou-process.md into claim-constant-sgd-near-a-loss-minimum-is-an-ornstein-uhlenbeck-process and the cross-domain bridge observation-mean-reversion-to-an-optimum-recurs-across-fossil-stasis-bonds-and-sgd. This is the load-bearing mechanism of the whole fossils ↔ bonds ↔ SGD bridge, and the capture itself flagged it: "the load-bearing 'SGD-as-OU' mechanism rests on one Tier-1 paper's analysis and should be double-checked before publication." The note is Tier 1 but kept seedling until this is discharged.

Source to read: Stephan Mandt, Matthew D. Hoffman, David M. Blei, "Stochastic Gradient Descent as Approximate Bayesian Inference," JMLR 18 (2017); arXiv:1704.04289.

Confirm verbatim the recorded quote — "Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution" — and its location.

Then establish the mechanism and its scope:

1. That the paper explicitly approximates the constant-SGD Markov chain by a continuous-time Ornstein–Uhlenbeck process (not merely a generic Gaussian stationary distribution).
2. The exact assumptions the OU approximation rests on — a constant (non-decaying) learning rate, a locally quadratic loss near the minimum, and gradient noise treated as (roughly) Gaussian with a fixed covariance. Note which are stated, which are idealizations.
3. The scope: the OU picture is a local description valid near a minimum, not a global account of SGD dynamics. How much of the "SGD is OU" slogan survives outside that regime?

Why it matters: if the OU approximation is faithful and its limits are stated honestly, the cross-domain bridge (paleoTS stasis, Vasicek rates, SGD) is real and publishable. If it is a looser analogy than the capture implied, the bridge note should be softened. High priority — this single verification gates whether the bridge is "exact, not analogical" or a structural echo.