Incremental Dimension Creation in Spatial Collective Choice

Spatial social choice usually asks whether a completed preferenceprofile admits a Euclidean representation. This paper fixes an inheritedcardinal spatial embedding and electorate, allows only the appending ofnew verified coordinates, and asks which ranking and majoritytransitions that operation can produce. The spatial constraint bites: anappended coordinate is generated by distances to one common alternativecoordinate and one voter ideal, so it cannot assign a free number toevery voter, alternative, and criterion triple. One appended coordinate makes each voter's scores affine in the newideal, giving an exact ranking-interval test and an allowable sequencewith at most C(m,2)+1 cells. Making a named alternative a Condorcetwinner has an exact coalition-interval characterization with O(nm)robustness tests and an exact distance-to-loss. Free-ideal named-winnerreachability is strongly NP-complete, and stays hard even on the unitinterval, though a bounded-cell dynamic program is polynomial for anyfixed number of unequal-level opponents. Under voter-homogeneous marginswith an odd electorate, the problem is linear-time, one coordinatepreserves majority transitivity, and two coordinates are necessary andsufficient to create a majority cycle. Minimum appended dimension is theminimum feasible rank of a positive-semidefinite Gram matrix, identifiedto within one coordinate by ordinal response rank.