Finitely many eigenvalues determine the signature of a hyperbolic orbifold

The Laplace spectrum of a closed orientable hyperbolic 2-orbifold determines its signature, the genus and the cone orders, but a measurement or a computation yields only finitely many eigenvalues, each to some accuracy. Among orbifolds with area at most A, systole at least ε and cone orders at most M, the first N eigenvalues determine the signature if each is known to within δ and no eigenvalue is omitted, with N and δ explicit and also computable by a certified procedure that always terminates. The general formulas are very large; with the gaps between signature terms certified in ball arithmetic, 10² to 10⁴ eigenvalues, known to within about 6 × 10⁻⁴ to 2.5 × 10⁻⁶, suffice for the classes of our computed examples. The bounds on the orders and on the area cannot be dropped, and at fixed accuracy N must grow at least like log M / log log M. Whether the systole bound is needed is open. The certified instances presuppose complete lists of eigenvalue enclosures, which our computed spectra do not provide; an a posteriori version of the test, run on ten computed spectra, gives consistency checks, not determinations. Code and data: https://doi.org/10.5281/zenodo.23740726. Companion paper: “How many heat invariants determine a hyperbolic orbifold?”, https://doi.org/10.5281/zenodo.23741037.