Marginally Correct Tool Caches Can Reverse Group-Normalized Policy Updates

Tool-result caching reduces repeated execution in agent training, but also couples rollout randomness. We study a two-action model in which independent and shared execution preserve every rollout's conditional reward distribution. Despite this marginal agreement, sharing one stochastic result per group can reverse the expected group-normalized policy update. We derive an exact finite-group expression: against a constant alternative, the shared update follows the probability of winning minus the probability of losing, rather than the difference in expected reward. A Bernoulli specialization yields a wrong-direction region and a non-vanishing update-variance floor as group size grows. Centering without group standard-deviation scaling preserves the expected-return direction in this model, using an existing estimator control. Exhaustive finite sums verify 540 configurations and 3,240 estimator evaluations, with a separate ordered-sequence checker. An implementation audit reproduces the sharing path in a pinned, unmodified TVCache stack using 256 scripted rollouts. These results do not measure language-model training performance or refute TVCache's deterministic-output contract. They establish that marginal output validity alone cannot certify a stochastic cache as training-equivalent.