Rungs, Statistics, and Macro Choice: Exact Conditions for Causal Control in Stochastic Hierarchies

Small changes in microscopic event laws can alter a macroscopic output, but monotonicity alone does not make that implication strict, a threshold does not create information, and an observed association does not establish control. This paper gives a corrected rung framework separating those issues. At a fixed context, a binary intervention changes the law of a microscopic state, which is then passed through deterministic or stochastic rungs. For a monotone Boolean output, stochastic domination gives a nonnegative macro effect. The effect is strict exactly when a monotone coupling crosses a reachable pivotal set. In the independent Bernoulli case, the effect is the line integral of changed input probabilities against their Boolean influences. Exact majority and drift-diffusion formulas quantify why response slopes can be largest near a threshold. A data-processing theorem simultaneously shows that downstream rungs cannot create information about the intervention, even when a powered device amplifies an observable. Its quantitative refinement bounds surviving total-variation contrast by the product of the rungs' Dobrushin contraction coefficients. A record-channel identity then states exactly when a macro outcome leaves a persistent downstream contrast. These results establish conditions for operational causal control, not a proof of metaphysical free will. Quantum mechanics is optional: a completely positive map may supply the micro channel, but classical thermal, chemical, or electronic noise obeys the same rung theorems. The framework yields a concrete experimental protocol for separating intervention, pivotal propagation, threshold gain, record writing, and endogenous policy selection.