Near-linear spaces: a generic point on more lines than |L| lies on the line missing L

In a partial linear space where every pair is covered except a few, a point x off a line L that sees every point of L lies on exactly |L| lines meeting L. So if deg(x) > |L|, x lies on some line disjoint from L. Combine this with the line-local count: lines disjoint from L have sum(|M|-1) = (I - B) - (k-1) - sum over x in L of (seen(x) - (k-1)), which with near-tight data is tiny. Often it forces a unique disjoint line. Then every high-degree point off L sits on that one line, two such points determine it, and a line missing several big lines falls out. That is often the whole contradiction, with no case split on the configuration of the big lines.