The Inaudible Exchange - Detecting Symmetry in Complementary Dirichlet-Neumann Spectra

Let the boundary of the unit disc be divided into alternating Dirichlet and Neumann arcs, and let complementation interchange the two boundary laws while fixing the switching points. For the mixed Dirichlet-Neumann (Zaremba) problem with four switching points, we prove rigidity: the complementary Laplacians are isospectral with multiplicities if and only if a disc isometry exchanges the Dirichlet and Neumann parts. This settles the four-switch case of a conjecture of Jakobson, Levitin, Nadirashvili and Polterovich. No genericity or nonvanishing-Fourier hypothesis is imposed. The proof connects heat invariants, finite Fourier data, switching-ray amplitudes and wave traces. Boundary balance follows from a signed heat-kernel measure; finite unoriented orbit separation recovers finite-jump circle functions up to O(2). We construct an all-orders regular transfer law at a smooth mixed-boundary switch and establish causal propagation for scalar trace extraction. We develop a globally clean closed-network calculus without assuming clean partial compositions. Closure removes open excess from the order while retaining its phase modulo eight; a two-state involution identifies coefficients surviving complementation. Applied to shortest-chord geometry, this yields a weighted spectral invariant; on the circle it becomes shortest-arc balance. For four switches, it equals the colour-exchange locus and the zero set of a cubic Fourier invariant. Intermediate results apply independently to finite-jump signals, gapped radial waves and finite two-state scattering networks.