A Zero Modular Entry Can Hide an Invertible Non-Scalar Braid
Sector: Kitaev (non-Abelian quantum doubles / mixed-sector readout)
Artifacts: research/kitaev/s3-charge-flux-monodromy-versus-modular-readout.md,
its exact checker, and research/kitaev/results/s3-charge-flux-monodromy.json.
Claim
For a pure charge representation pi transported around a conjugacy-class
flux space C, the full monodromy operator is
M_(pi,C) = direct_sum_(g in C) pi(g),
and its scalar modular readout is only the trace:
tr M_(a,b) = 6 S_(a,b).
For the two-dimensional standard charge C around transposition flux D or
E, exact calculation gives
characteristic polynomial = (x-1)^3 (x+1)^3
minimal polynomial = x^2-1
trace = 0
determinant = -1.
Thus S_CD=S_CE=0, while the underlying six-dimensional braid is invertible
and non-scalar. The zero is cancellation between three positive and three
negative reflection eigenvalues, not absence of braid response.
For the same charge around three-cycle flux, the minimal polynomial is
x^2+x+1, the characteristic polynomial is (x^2+x+1)^2, and the trace is
-2=6(-1/3). Operator spectra therefore distinguish transposition and
three-cycle flux families.
Pure-charge monodromy nevertheless has a capability kernel: it acts on flux
labels but not on the centralizer irrep, so it gives the same operator for
D/E and likewise for F/G/H. Resolving those anyons requires magnetic or
dyonic probe coefficients.
The checker constructs the standard S3 representation from generators,
verifies the group law, audits all six charge/family pairs, proves
invertibility and the minimal polynomials, and verifies tr M=6S. Seven
aggregate gates pass; fresh stdout matches the saved JSON.
Boundary
Full monodromy is not an elementary exchange map. This entry does not choose
mixed fusion bases, resolve centralizer irreps, derive channel-wise mixed R
symbols, or verify mixed hexagons. Equality under this probe is a readout
kernel, not equality of anyon sectors.