How many heat invariants determine a hyperbolic orbifold?

In a short time t, heat on a closed hyperbolic 2-orbifold with cone points feels, up to an error smaller than every power of t, only the geometry near each point, and the coefficients c₁, c₂, … of the short-time expansion of the heat trace, the heat invariants, record what it has felt at each order in t. We ask how many orders are needed to determine the genus and the cone orders. If every cone order is at most M, the first M + 1 heat invariants suffice at every area, and no smaller number works at all areas. Without that bound the number needed at area at most A grows only through large cone orders, at least like √A and at most linearly, and a lower bound cA^α is equivalent to the bound O(k^(1/α)) for the least size of a degree-k Prouhet–Tarry–Escott solution. In variable curvature no finite number suffices. Four heat invariants determine the signature of a triangle orbifold, and three determine a triangle orbifold among triangle orbifolds. The least-area pair sharing two invariants, O(2,8,8) and O(3,3,12), is isolated among triangle orbifolds because the cubic carrying such collisions has rank 0; with variable curvature it carries metrics whose heat invariants agree to all orders. This record contains the manuscript and its electronic supplementary material (Online Resource 1). Code and data: https://doi.org/10.5281/zenodo.23740674. Companion papers: “Finitely many eigenvalues determine the signature of a hyperbolic orbifold”, https://doi.org/10.5281/zenodo.23741059; “Triples with equal sum and equal reciprocal sum”, https://doi.org/10.5281/zenodo.23741067.