= Area separation of convergent Sunada families
{title2=$\operatorname{area}(h_k)=\operatorname{area}(h_0)(1+2^{-k})$}
Suppose a fixed pair of <Gassmann equivalent> finite covers of a compact two-dimensional orbifold has isometric limiting hyperbolic metrics. Choose nearby orbifold-smooth <nowhere locally homogeneous metrics> and rescale them to the displayed distinct areas. Scaling by a positive constant multiplies area by that constant in dimension two, preserves local rigidity, and preserves convergence when the constants tend to one. <Sunada theorem> supplies equal spectra within each pair. Local rigidity forces any quotient isometry to descend to the identity on the base; nonconjugate covering subgroups therefore exclude within-pair isometry. Distinct areas exclude all cross-pair isometries and spectra, because the leading <heat trace> coefficient is area divided by $4\pi t$. Fixed markings of the two isometric limiting covers give convergence to the same metric. The existence of those isometric limits is an additional geometric input, not a consequence of <Gassmann equivalence> alone.
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