Symmetric unit-norm tight frames for three worst-case erasures: magic sizes, irregular optima, and an average-case check

Let G be a real unit-norm tight frame (FUNTF) used to encode a vector, and suppose that any s = n − d coefficients may be lost. The worst-case condition number of the surviving d × d submatrix is the standard stability measure in coded distributed computation. Through the Naimark complement the problem reduces to n lines in R^s. For s = 3 we report, with coordinates and two independent verifications, frames whose worst-case condition number is 2.0, 2.5 and 4.0 times smaller than the best real harmonic frame for n = 12, 24, 60. They are orbits of a single vector under the rotation groups of the tetrahedron, cube and icosahedron (a known construction of group frames), and in all three cases unconstrained local optimization could not improve them. For n = 16 and 20, where no such group is available, the best frames we found are irregular (no nontrivial symmetry) and improve the previous values by 1% and 12%. A scaling relation κ ≈ 0.147 n², fitted on the symmetric sizes, predicted values for n = 16, 20 that were not confirmed. In the average case over all erasure patterns, the symmetric frames give no relevant advantage over harmonic frames in fp16 or int8 arithmetic. We do not claim optimality of any frame.