Uncomputable finite-size gap exponents

We classify the nonnegative finite-size spectral-gap exponents of translation-invariant nearest-neighbour Hamiltonians on the square lattice, with finite local dimension, integer interaction matrices and open boundaries. The gap is measured above the entire ground space on an $L\times L$ box. The existing exponents in $\Delta(L)=L^{-z+o(1)}$ are exactly the nonnegative limit-computable ($\Delta^0_2$) reals. The finite nonnegative lower/upper exponent pairs are exactly the ordered pairs consisting of a left-$\Sigma^0_2$ real and a left-$\Pi^0_2$ real. In particular, one fixed integer-valued model has exponent equal to the halting probability of any chosen prefix-universal machine. A classical boundary preference fixes a size counter and a computation tableau; a reversible mixed-radix counter supplies the small gap. We bound every sector of the full Hilbert space. The closing modes live on a boundary segment of length $O(\log L)$, and the local dimension may depend on the prescribed pair. These are finite-box exponents, with no assertion about bulk dispersion or stability of thermodynamic critical exponents.