Counting Quadratic Presentations over Finite Fields : A Growing-Ball Koszul Density Theorem

Fix a prime $p$. We count all ordered quadratic presentations with at most $B$ generators, at most $n^2$ relations on $n$ generators, and coefficients in a chosen field of order $p^e$, $1\\le e\\le B$. With uniform counting on this finite disjoint union, the proportion defining Koszul algebras tends to one. More precisely, the failure probability is at most $(2B^4+B^2)/p^B$. The mechanism is explicit: invertible relation matrices in the largest stratum already have density one, and their algebras are square-zero extensions. The exact count of invertible matrices sharpens this: the proportion of presentations whose algebra is not such a square-zero extension lies between $1/(2p^B)$ and $3/p^B$ for $B\\ge2$ and is asymptotic to $p^{-B}$. This is a theorem about a specified measure on presentations, not about each fixed relation dimension or about uniformly counted isomorphism classes. Indeed, for two generators and two random relations we prove that Koszul probability tends to zero as the field grows. The same bound holds, with the same proof, for coefficients drawn from finite boxes of $c^e$ elements in arbitrary fields, which is the form the statement takes over infinite fields; boxes of polynomial size $e^k$ with $k\\ge5$ suffice.