  We constrain the low-energy spectra of Laplace operators on closed hyperbolic
manifolds and orbifolds in three dimensions, including the standard
Laplace-Beltrami operator on functions and the Laplacian on powers of the
cotangent bundle. Our approach employs linear programming techniques to derive
rigorous bounds by leveraging two types of spectral identities. The first type,
inspired by the conformal bootstrap, arises from the consistency of the
spectral decomposition of the product of Laplace eigensections, and involves
the Laplacian spectra as well as integrals of triple products of eigensections.
We formulate these conditions in the language of representation theory of
$\mathrm{PSL}_2(\mathbb{C})$ and use them to prove upper bounds on the first
and second Laplacian eigenvalues. The second type of spectral identities
follows from the Selberg trace formula. We use them to find upper bounds on the
spectral gap of the Laplace-Beltrami operator on hyperbolic 3-orbifolds, as
well as on the systole length of hyperbolic 3-manifolds, as a function of the
volume. Further, we prove that the spectral gap $\lambda_1$ of the
Laplace-Beltrami operator on all closed hyperbolic 3-manifolds satisfies
$\lambda_1 < 47.32$. Along the way, we use the trace formula to estimate the
low-energy spectra of a large set of example orbifolds and compare them with
our general bounds, finding that the bounds are nearly sharp in several cases.
