Hearing the Sides: Recovering a Planar Rectangle from Eigenvalues

We present a direct, index-free method to recover the side lengths of a planar rectangle the spectrum of its Dirichelet Laplacian, assuming only access to a finite subset of eigenvalues. No modal indices $(m,n)$ are available, and the list may begin at an arbitrary unknown offset; in particular, the lowest eigenvalues may be missing, so classical formulas based on $\lambda_{1,0}$ and $\lambda_{0,1}$ cannot be used. Our reconstruction procedure extracts geometric information solely from the asymptotic density and oscillatory structure of the ordered spectrum. The area $ab$ is obtained from the high-frequency Weyl slope, while the fundamental lengths $2a$ and $2b$ appear as dominant periodic--orbit contributions in the Fourier transform of the spectral fluctuations. This separation of smooth and oscillatory components yields a robust, offset-agnostic recovery of both side lengths. The result is a fully index-free algorithm that reconstructs the geometry of a rectangular planar domain even when the spectrum is incomplete and all modal information is lost.