Quota and population monotonicity across house sizes are incompatible for apportionment to four states

Abstract

Apportionment converts fractional entitlements into integer seat allocations. Quota requires each state to receive the floor or ceiling of its standard quota, whereas population monotonicity prohibits a state whose population weakly increases from losing a seat to a state whose population weakly decreases. Gölz, Peters, and Procaccia recently removed the order-preservation assumptions used in classical incompatibility results by giving a five-state construction; together with the three-state Webster possibility result, this left four states as the unresolved boundary. We prove that no deterministic four-state apportionment solution satisfies both axioms under their definition, which permits the compared house sizes to differ. The proof is a finite logical gadget. Conditional on one quota choice at a central profile, twelve auxiliary profiles encode three bits and force a frustrated cycle. The argument uses only transfers between states whose populations are unchanged and assumes neither anonymity, neutrality, order preservation, homogeneity, nor other regularity conditions. It also applies to relative population monotonicity. Geometrically, the result is a global compatibility obstruction for quota-constrained lattice rounding, rather than an average-distortion bound.