What crossing symmetry forbids just above $c=1$

What does a unitary two-dimensional conformal field theory look like just above $c=1$? Using the numerical conformal bootstrap, we study the four-point function of a scalar Virasoro primary $ϕ$ such that no primary lighter than $ϕ$ appears in its operator product expansion with itself. Slightly above $c=1$, crossing symmetry excludes the interval 0.17Δ_ϕ1, although the compact boson realizes every such Δ_ϕ at $c=1$. Combining this exclusion with results from the modular bootstrap, we show that the lightest primary of a compact unitary conformal field theory at $c=1.0005$ or $1.001$ has dimension Δgap<0.17. Further from $c=1$, the largest excluded central charge at fixed Δ_ϕ has two sharp minima near Δ_ϕ=1/2 and $4/7$. At these values it lies within 2×10⁻⁷ of $c=8/7$, where the Z₅ parafermion and its charge-conjugation orbifold have scalars of exactly these dimensions.