Exact Spectra of Generalized Cubic Subdivision Matrices

Abstract

The spectrum of a subdivision matrix controls local convergence near an extraordinary mesh element. For every surface valence n ≥ 3, Dietz's generalized cubic B-spline construction produces an initial-element matrix

S_n = exp((log 2)(L̃_n − I)) exp((log 2)(R̃_n − I)),

and a released double-ring matrix Ŝ_n assembled from S_n and regular cubic stencils. The source established two independent 1/2-eigenvectors but left open whether 1/2 is subdominant, has exactly the expected multiplicity, and is semisimple. We resolve all three questions and give the complete spectra of both released surface matrices. A discrete Fourier transform reduces S_n to one 3 × 3 block and n − 1 blocks of size 2 × 2; a phase-free block-triangular form gives the outer spectrum

{0^[24n], (1/64)^[n], (1/32)^[n], (1/16)^[n], (1/8)^[n]},

where brackets denote algebraic multiplicity. Thus both matrices have a simple dominant eigenvalue 1, a semisimple subdominant eigenvalue 1/2 of algebraic and geometric multiplicity two, and no other eigenvalue of modulus at least 1/2. For n ≥ 5, the subsubdominant eigenvalue satisfies

1/2 − 3π²/(4n²) + π⁴/(2n⁴) + O(n⁻⁶)

as n → ∞, explaining the vanishing spectral gap. A proved tensor identity gives the expected multiplicity three for the semi-regular initial-volume family. The analogous double-shell conclusion remains conditional on the tensor identity stated with a proof sketch in Dietz's source. Exact-identity checks, independent reconstructions, and Lean certificates accompany the paper. Fully irregular three-dimensional matrices and a complete C¹ theorem are not covered.