A sharp stability transition for low-rank tomography with four missing Weyl coefficients

Let d=9n with arbitrary integer n>=1, and omit the four cyclic Weyl coefficients associated with X_d^(d/3), its adjoint, Z_d^(d/3) and its adjoint. We determine the exact inverse Frobenius constant on differences of density matrices of rank at most r. Writing x=r/n, the squared fraction of energy that can be hidden in the missing coefficients is 2x/3 for 0<x<=1, and (x+sqrt(x^2-3x+3))/3 for 1<=x<2. The resulting inverse constant is attained by explicit orthogonally supported states and diverges as sqrt(2n/(2n-r)) at the exact nonidentifiability threshold r=2n. The same threshold holds when a rank-r state must be distinguished from every positive state of unrestricted rank. For this stronger task we give a separate, conservative stability bound and a convex least-squares recovery consequence. The proof reduces the missing-coefficient space to additive 3 by 3 spectral grids, then solves two four-dimensional symmetric matrix pencils. These are results for one fixed erasure pattern, not for arbitrary four erasures or a measurement scheme with subquadratically many coefficients.