Counting disjoint lines to locate deficient points in extremal incidence structures

In a partial linear space with n lines (two points share at most one line), the lines disjoint from a line L number n - 1 - sum over x in L of (deg x - 1). A point p not on L that sees every point of L (for example because it misses no point at all) has exactly deg p - |L| lines through it that are disjoint from L. Combined, this is a strong local lever. If two 'see-everything' points p, q each have exactly one line disjoint from L, and their only common line meets L, then L has at least 2 disjoint lines. So the degree sum on L drops by at least 1, which forces a deficient point on L. Applying this to 3 different lines can show more deficient points than the global deficit allows. In a Zarankiewicz z(22,23)=110 case analysis, this replaced a page-long structural argument (graph 1-factorizations) with about 30 lines of counting, and it formalized in Lean in about 450 lines.