Verified Criterion Expansion and Collective Choice: Unanimity Cycles and the Geometry of Dimension-Proofness

Political actors can change the outcome of a collective decision without changing the options, misreporting anyone's preferences, or adding new candidates: they can change what the options are understood to be about, by introducing a new criterion along which the same fixed alternatives are suddenly evaluated, so that a proposal once treated as a distributional bargain is recast as a matter of security, constitutional principle, regional identity, or administrative competence. William Riker called this the manipulation of dimensions and thought it the most powerful tool of political strategy, and this paper gives it a formal model. Novelty is treated as relative to an institution: the new criterion already exists in the modeler's language but not yet in the institution's current working vocabulary, the same device used in theories of growing awareness. A fresh criterion carries a publicly verifiable measurement of the alternatives, but different voters may respond to it differently, and the strategist who decides which criterion to admit has a goal fixed in advance, before the situation and the criterion are chosen. The word manipulation here is procedural rather than a welfare judgment, since the criterion is genuine and the voters' reactions sincere and a rule may be right to change its outcome; what the paper isolates is the adversarial power to select a criterion and a tension between a rule's responsiveness and its robustness to such selection, shown by an explicit case that moves the winner from one Condorcet winner, the option that beats every other in pairwise majority voting, to a different one. The results form a graded hierarchy rather than a single impossibility. Every way of admitting criteria induces a directed graph on preference profiles, where one profile leads to another when some admissible new criterion can carry the first into the second, and a rule is immune to dimension creation for a fixed objective exactly when its outcomes are constant on each cluster of mutually reachable profiles and never improve along the direction of reachability; an onto immune rule exists precisely when the number of clusters is at least the number of alternatives, and any further requirement that fixes outcomes on some profiles is compatible with immunity only when those forced outcomes respect reachability, since any cycle among them defeats every rule obeying the requirement. The sharp consequences are that if responses to even a single pair of alternatives can be pushed in opposite directions a cycle appears among unanimous profiles, so no unanimous, Pareto-efficient, or Condorcet-consistent rule is immune; if responses to every pair can be pushed both ways only rules that ignore all preferences survive; and if admissible criteria only ever push one way a rule can be nonconstant, onto, strategyproof, and immune at once. The paper derives this graph from underlying attribute data, counts the clusters exactly through acyclic orientations of a comparability graph, and in a clean binary case establishes a chain-versus-antichain threshold in which two overlapping criteria already break unanimity while a classical combinatorial result, Sperner's theorem, fixes how many criteria force complete constancy; a closing corollary ties the framework to the classical strategyproofness impossibility, showing that a strategyproof rule survives dimension creation exactly when its hidden decisive voter has no available criterion that serves the strategist's objective.