UnitScope: Limits and Conditions for Bit-Exact Unit Transport in Stateful Safe RL

Changing the numerical unit of a safety cost leaves a constrained Markov decision process unchanged, but it can change the floating-point program used to train the policy. We study this problem in four tiers. First, real-arithmetic theorems show that scalar clipping and shared vanilla-Adam groups cannot serve blocks with incompatible unit signatures. Second, our frozen preregistered study at the non-dyadic scales $\alpha\in\{0.1,10\}$ found that finite-tolerance one-update certificates did not establish closed-loop equivalence: 0/4 Holm-adjusted primary tests and 0/32 corrected-margin sensitivity decisions passed, and the release status remains NOT RELEASED. Third, we prove a post-study theorem: dyadic transport through a fixed deterministic typed graph is bit-exact whenever every rounding stays range-safe, and an ideal-format converse shows that no non-dyadic scale can act exactly on every representable value. Captured replays agree with the theorem: float64 passed 144/144 dyadic cells, float32 passed 87/144, and every failure was a subnormal crossing or a mismatched exponent-zero graph. Fourth, after separately preregistered adapter-side repairs, $\alpha=2$ float32 training on a pinned one-worker CPU stack (PointGoal, two seeds per library) stayed bit-exact with its reference for 60,000 steps per branch, well past the region where the frozen study first diverged. A silent range monitor certifies the normal-range premise only for observed registered operands whose rounded values are nonzero; signed-zero fixed points are recorded but not certified, and alerts are inconclusive. We therefore report a scoped existence result for repaired adapters, not an unconditional floating-point symmetry and not a behavior-preserving release.