This paper introduces a new structural lemma for strategy-proof social choice, the Mutual Exclusion of Influence, and develops three layers of consequences connecting pairwise control, Boolean influence theory, and information-theoretic entropy. The Mutual Exclusion of Influence theorem states that for any strategy-proof social choice function, two distinct voters cannot simultaneously have pairwise influence over the same alternative pair at the same profile while ranking that pair in opposite directions. Shared control of a pair is therefore incompatible with disagreement on that pair. The proof is profile-local and operates through an adjacency bottleneck: by placing two alternatives adjacent in one voter's report, a strategy-proof constraint forces the crossed-profile outcome to be one of those two alternatives, contradicting the derivation that it must be neither. A fiber decomposition of shared control follows, showing that for any pair and two voters at a fixed context, at least one of four exclusive regimes holds: voter i never enables voter j to control the pair, voter j never enables voter i, shared control is possible only under mutual agreement that a is preferred, or shared control is possible only under mutual agreement that b is preferred. The first two regimes are exclusive control; the latter two constitute the forced-agreement regime permitted by mutual exclusion. The paper also proves a local pivot lemma: along an adjacent transition profile sequence where voter k moves b from second to first while a moves from first to second, if the outcome switches away from a it must switch precisely to b. The paper gives an explicit example showing why this adjacency repair does not by itself yield a decisive voter, because mutual exclusion says nothing once other voters agree with the pivot voter or option sets change during the modification. Global results for realized range at least three are therefore obtained by combining range projection with the Gibbard-Satterthwaite theorem rather than from mutual exclusion alone. The binary regime is developed systematically. A range projection theorem shows that any strategy-proof function factors through the restriction of preferences to its realized range, yielding an onto strategy-proof function on that range. The realized-range trichotomy follows: range of size one gives a constant, range of size two gives a monotone committee rule on the pair, and range of size three or more gives a dictatorship on the realized range. The monotone Boolean reduction for binary range is proved, establishing a bijection between strategy-proof rules on a pair and monotone Boolean functions. The product-prior entropy law is the central analytic result. For any monotone Boolean function with independent Bernoulli inputs, the conditional entropy of the output given all inputs except coordinate i equals the binary entropy of the i-th marginal times the pivotal probability of that coordinate. This is an exact identity, proved by a two-case argument conditioning on the pivotality event. Summing over all coordinates yields an entropy-influence inequality stating that the weighted sum of pivotal probabilities at least equals the output entropy, with equality if and only if the function is constant or a coordinate projection. The proof proceeds by induction on the number of variables, with the inductive step using a weighted scalar entropy lemma proved via a coupling argument involving conditional mutual information. This lifting passes through a lemma showing that the ranking-prior conditional entropy of a binary social choice function equals the Boolean conditional entropy of its monotone reduction, under any product ranking prior. Under impartial culture the product-prior identity reduces to the exact identity between the conditional entropy and the Boolean influence of each coordinate, and the entropy-influence inequality recovers the total influence bound. A rigidity theorem proved via Ellis's stability theorem for sets of small edge-boundary excess shows that any monotone Boolean function with total influence within epsilon of the output entropy must be close in disagreement probability to either a constant or a coordinate projection. Exact formulas for anonymous threshold rules under impartial culture are derived, including the pivotal probability, the output entropy, the total influence, and the central-quota maximizers, with the majority rule achieving asymptotic total influence of order the square root of two n over pi. In the entropy framework developed for the third part, the conditional entropy of a strategy-proof function's output given all inputs except voter i is proved to equal the logarithm base two of the size of voter i's option set, exactly and pointwise in the context. This implies that a voter with maximal average influence entropy is a dictator on the realized range. For realized range at least three, entropy concentrates entirely on the dictator. The synergy obstruction for two voters characterizes negative interaction information completely. Interaction information is proved to equal minus the conditional mutual information between the two voters' rankings given the outcome, making it non-positive. The two-voter synergy characterization theorem proves that the following four conditions are equivalent: interaction information is strictly negative; the two voters' rankings are not conditionally independent given the outcome; the realized range has exactly two elements and the monotone Boolean reduction is AND or OR; and some pair is in the forced-agreement regime of the fiber decomposition. Range locking follows: if some pair is in the forced-agreement regime for two voters, the realized range has exactly two elements. A cross-ownership impossibility theorem proves that if one voter exclusively owns the pair of a and b in the sense of controlling it without the other voter ever controlling it, the other voter cannot exclusively own the pair of a and c for any third alternative in the realized range. The paper then recovers the complete classification of the 17 strategy-proof social choice functions for two voters and three alternatives as a corollary of the realized-range trichotomy and the two-bit monotone classification. In the single-peaked domain, a cut-geometry theorem applies the product-prior entropy law to each cut of a generalized median voter scheme by representing the cut output as a threshold function of independent peak-position bits, deriving the exact per-cut conditional entropy formula, and bounding the total output entropy via subadditivity.
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