Mean-reversion toward an optimum — the Ornstein–Uhlenbeck process — recurs independently across fossil stasis, bond pricing, and SGD near a loss minimum

Three separately-developed models, in three fields that share almost nothing else, reach for the same stochastic object — the Ornstein–Uhlenbeck (OU) process, a mean-reverting diffusion with a stationary Gaussian distribution around a fixed point:

• Fossil stasis. Quantitative paleobiology fits an OU model to trait time-series, treating morphological stasis as a trait "pulled towards the optimum" near a fixed peak in the adaptive landscape (claim-ou-model-recasts-stasis-as-active-mean-reversion-to-an-optimum, within the model-selection framework of claim-paleots-fits-fossil-stasis-as-a-selected-stochastic-model).
• Bond pricing. The Vasicek interest-rate model is an OU process, mean-reverting toward a long-run rate — the same equation the process took from a Brownian particle decelerating under friction (claim-ornstein-uhlenbeck-process-links-brownian-motion-and-the-vasicek-model).
• Machine learning. Mandt, Hoffman and Blei approximate constant-learning-rate SGD near a loss minimum as a continuous-time OU process, mean-reverting in the quadratic well around the minimum (claim-constant-sgd-near-a-loss-minimum-is-an-ornstein-uhlenbeck-process).

The shared structure is a restoring force toward the bottom of a well: a fitness peak, a price equilibrium, a loss basin, each stabilized by a pull whose strength is a fitted parameter. This connects two vault clusters that had no prior link — the punctuated-equilibrium/stasis notes (claim-punctuated-equilibriums-novel-addition-was-stasis-emphasis, claim-red-queen-hypothesis-names-running-to-stay-in-place) and the ML-optimization notes (claim-robbins-monro-1951-stochastic-approximation, claim-amari-1998-natural-gradient-fisher-steepest-descent, backpropagation-gap).

Whether this is one shared mathematical object or three structurally-similar analogies is left open, not asserted — the same open question the vault already holds for gradient geometry (question-gradient-geometry-one-object-or-three-analogies). The recurrence is recorded as an observation; the "same equation, exact bridge" reading is confined to the commentary below.