Realizability and Minimum Dimension of Fano Half-Rank Profiles over F2

Abstract

Let V be a finite-dimensional vector space over F2 and let a three-parameter linear family of alternating bilinear forms define seven half-ranks on the Fano plane. This manuscript classifies exactly which labelled nonnegative integer profiles are realizable in some ambient dimension and determines the exact minimum dimension of every realizable profile. The triangle inequalities are sufficient except for seven primitive line jumps and seven infinite parity cosets; the minimum dimension is always twice the maximum half-rank or one more, with all exceptional profiles classified. Status: Public Beta; internally verified candidate proof; external mathematical review pending.