Source: mpeac, 'Why aren't the optimal entry/exit thresholds for OU pairs trading relatively invariant to shifts in the OU mean?', quant.stackexchange.com/questions/72234 (2022-09-18), which discusses Leung & Li, 'Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit'. Classification: (a). The post is a theory question, but it names a concrete mechanism: trade a mean-reverting OU spread, entering at a cost-aware threshold below/above the mean and exiting at an optimal level, with a stop. Answer to the question, which shapes the design: Leung-Li's thresholds are not translation-invariant because their problem is long-only optimal stopping on the discounted LEVEL of the spread (the reward is e^{-rt}(X_t - c)). Discounting a level that changes sign makes the value depend on where theta sits relative to 0. Theirs is not a symmetric long/short problem on the deviation X - theta. For a two-sided spread trader with negligible carry, the right formulation is in deviation units: entry and exit bands are set relative to a point-in-time theta and are symmetric for both sides. That is the invariance the asker expected. We use that formulation. The open question is which spread really is OU. Statistical pairs are dead in this factory (L113, the mean_reversion tag is 0/203), so we use the one spread class where mean reversion is contractual: two share classes of the SAME company. They have identical cash flows and differ only in voting rights. The log price ratio is pinned to a slowly drifting voting premium (theta), and liquidity and index-flow shocks push it away from that premium. Fit OU on log(A/B) over a rolling window, trade z-deviations from the rolling theta, and hold for days. Dollar-neutral: long the cheap class, short the rich class, same notional. All three pairs run in one strategy instance, each as its own independent sleeve. USEQ is under-represented at 5.0% of experiments, and long_short is under-represented at 15.7%. PRE-STUDY (point-in-time rolling mean and std, entry |z|>k, exit at z crossing 0 or a 20-session cap, daily closes from the catalog via load_normalised). Gross return per trade is in % of total two-leg notional. GOOGL/GOOG since 2014-04: L60/k2.0 gives 83 trades, gross +0.12%, win 77%, 12 of 13 years positive. L20/k2.0 gives 138 trades, +0.12%, win 85%, 13 of 13 years positive. NWSA/NWS since 2013-07: L60/k1.5 gives 126 trades, +0.25%, median +0.43%, win 72%. L120/k2.0 gives 57 trades, +0.26%. NWS has 4 losing years: 2013, 2021, 2023 and 2024. FOXA/FOX since 2019-03 is the weakest: L20/k2.0 gives 88 trades, +0.10%, win 76%, and the longer windows are about 0. FOX is kept as a third, independent sleeve so that the hypothesis can be checked against it rather than tuned to it.