\begin{abstract}
We consider modular-invariant torus partition functions of central
charge $c>1$ that have a unique vacuum, an expansion in non-degenerate
Virasoro characters with positive integer multiplicities, integer spins,
no nonvacuum state at the unitarity bound, and parity symmetry. We prove
that for every $g>0$ there is $\epsilon(g)>0$ such that no such
partition function with $1<c<1+\epsilon(g)$ has all nonvacuum primary
dimensions at least $g$. Thus the spectral gap closes as $c\to1^+$ in
this class, as suggested by the numerical bootstrap study of Benjamin,
Langhoff, Li and Thaler. The proof is by contradiction and compactness.
Using the crystalline-measure reduction of Eberhardt and Levine, we
show that every limit at $c=1$ is an equal-weight average of two
compact-boson partition functions, at least one at a half-integer
radius. Positive Gaussian test functions adapted to the two circle
lattices then force every state of a nearby partition function, through
a logarithmically growing energy window, to lie within a power of $c-1$
of a lattice point. On the other hand, the central-charge shift of the
vacuum contributes a second-order term at zero momentum that no
distribution supported on circle lattices and compatible with modular
invariance can carry; a finite family of Gaussian tests detects this.
Limits with two half-integer radii, where lattice points collide,
require a separate analysis. The threshold $\epsilon(g)$ is not
effective; we describe the additional estimates that would make it so.
\end{abstract}