Let the exact homogeneous hard-margin support vector machine be trained on
m independent observations from a Borel probability law on a real Hilbert space. We prove that, with score zero counted as an error, there is a universal numerical constant
C such that
\Pp(γm>0,\Risk(um)>mC(Km+logδ1))≤δ.
Here
γm is the empirical homogeneous margin,
um is the exact minimum-norm unit-margin separator,
rm is the largest training radius, and
Km:=rm2\normum2=rm2/γm2 on
{γm>0}. The proof is driven by a deterministic deletion problem. Given vectors
x1,…,xn in the unit ball, delete a set
B of constraints and let
uB be the closest point to the origin that satisfies every retained unit-margin constraint. Suppose that
\normuB2≤k and that every deleted vector has nonpositive score under
uB. We prove that a family of such deletion sets of cardinality
q has size at most
exp(8k+2q). The conceptual step is an exact identity obtained from the KKT representation of
uB. For a random deletion set, the identity converts the mean squared spread of the separators into a weighted sum of score deficits. It therefore forces a coordinate whose deletion status separates the two conditional means by a quantitatively large amount. Revealing that coordinate decreases the conditional separator variance enough to control the binary entropy of the split. An entropy induction gives the deletion count, and an exact factorial ghost-sample identity converts that count into the stated high-probability SVM bound.