  Finding discriminant functions of minimum risk binary classification systems
is a novel geometric locus problem -- which requires solving a system of
fundamental locus equations of binary classification -- subject to deep-seated
statistical laws. We show that a discriminant function of a minimum risk binary
classification system is the solution of a locus equation that represents the
geometric locus of the decision boundary of the system, wherein the
discriminant function is connected to the decision boundary by an exclusive
principal eigen-coordinate system -- at which point the discriminant function
is represented by a geometric locus of a novel principal eigenaxis --
structured as a dual locus of likelihood components and principal eigenaxis
components. We demonstrate that a minimum risk binary classification system
acts to jointly minimize its eigenenergy and risk by locating a point of
equilibrium, at which point critical minimum eigenenergies exhibited by the
system are symmetrically concentrated in such a manner that the novel principal
eigenaxis of the system exhibits symmetrical dimensions and densities, so that
counteracting and opposing forces and influences of the system are
symmetrically balanced with each other -- about the geometric center of the
locus of the novel principal eigenaxis -- whereon the statistical fulcrum of
the system is located. Thereby, a minimum risk binary classification system
satisfies a state of statistical equilibrium -- so that the total allowed
eigenenergy and the expected risk exhibited by the system are jointly minimized
within the decision space of the system -- at which point the system exhibits
the minimum probability of classification error.
