Inverse Spectral Geometry of Cut-and-Project Quasicrystals

We address the inverse spectral problem for cut-and-project quasicrystals: what hidden geometry does the perpendicular spectral hierarchy force? Papers 19-20 established the forward result — the lift-filter-compression mechanism produces monotonic perpendicular grading across spectral bands, verified at internal dimensions N=4,5,6. The present paper inverts this. Three layers of recoverable structure are identified. First, grading depth: the number of stable spectral bands determines the parent lattice dimension N. Second, window compatibility: monotonicity of ⟨|k⊥|⟩ diagnoses whether the acceptance window respects the point group. Third, symmetry family: stabilised band multiplicities (1, binom(N,1), …, binom(N,N), 1) determine the symmetry type as discrete integer data. The non-uniqueness of the inverse problem is made precise through spectral equivalence classes. We define a filtration of observable data — band count, multiplicities, grading values, irrep sub-decomposition, gap labels — and prove that the corresponding equivalence classes are nested. Eigenvalue interval ratios between resolved bands provide a concrete Level 3 diagnostic: for the icosahedral QC under three acceptance windows, compatible windows produce a non-trivial ratio (ρ₀ = 0.824), while incompatible windows produce equal spacing (cube: ρ₀ ≈ 1) or simple rational ratios (octahedron: ρ₀ ≈ 1/2), distinguishing window symmetry classes from spectral data alone. We conjecture that the filtration stabilises at a finite threshold n*, beyond which additional spectral data does not further constrain the recoverable geometry. Twenty-first paper in a series; companion papers at DOI 10.5281/zenodo.19422381, 10.5281/zenodo.19420222, 10.5281/zenodo.19393159.