Three Settings Identify Pure D(S3) Sectors, but Seven Are Needed for Mixtures
Sector: Kitaev (readout domains / mixture tomography) Artifacts: research/kitaev/s3-pure-sector-readout-versus-mixture-tomography.md, its checker, and results/s3-sector-mixture-tomography.json.
Claim
The normalized-monodromy family (Im twist,mu_D,mu_F) separates all eight pure D(S3) sector labels. It is not faithful on arbitrary classical mixtures of those sectors. With the normalization row included, its affine readout matrix has rank four and kernel dimension four.
Exact collisions include
1/2 A + 1/2 B =readout F
and
1/2 D + 1/2 E =readout 2/3 C + 1/3 F.
No amount of repeated sampling distinguishes either pair under the three-setting instrument.
On the frozen ten-real-setting surface
{Re twist, Im twist, mu_A, ..., mu_H},
arbitrary mixture tomography requires exactly seven settings. Dimensional rank forbids six or fewer, and exhaustive enumeration finds 21 seven-setting families of augmented rank eight. One witness is
(mu_B,mu_C,mu_D,mu_E,mu_F,mu_G,mu_H),
whose normalization-augmented determinant is -81.
Seven exact gates pass and fresh stdout matches the saved JSON.
Boundary
The theorem concerns classical mixtures of superselection sectors, not coherent superpositions of inequivalent charges or tomography inside an anyon internal space. The minimum is relative to real scalar settings and changes with multi-outcome effects, adaptive protocols, prior support, or a different cost model.