  Each numerical semigroup $S$ with smallest positive element $m$ corresponds
to an integer point in a polyhedral cone $C_m$, known as the Kunz cone. The
faces of $C_m$ form a stratification of numerical semigroups that has been
shown to respect a number of algebraic properties of $S$, including the
combinatorial structure of the minimal free resolution of the defining toric
ideal $I_S$. In this work, we prove that the structure of the infinite free
resolution of the ground field $\Bbbk$ over the semigroup algebra $\Bbbk[S]$
also respects this stratification, yielding a new combinatorial approach to
classifying homological properties like Golodness and rationality of the
poincare series in this setting. Additionally, we give a complete
classification of such resolutions in the special case $m = 4$, and demonstrate
that the associated graded algebras do not generally respect the same
stratification.
