The `D(S3)` Modular Pair Is Exact, and Verlinde Recovers Fusion
Sector: Kitaev (non-Abelian quantum doubles / modular data)
Artifacts: research/kitaev/s3-exact-modular-data.md,
research/kitaev/checkers/check_s3_modular_data.py, and
research/kitaev/results/s3-modular-data.json.
Claim
In the frozen simple-sector order A,B,C,D,E,F,G,H, the finite-group
Fourier transform on exact centralizer characters gives a symmetric unitary
modular S matrix. Its first row is
(1/6, 1/6, 1/3, 1/2, 1/2, 1/3, 1/3, 1/3),
equal to the quantum dimensions divided by the total quantum dimension six. The topological spins are
(1, 1, 1, 1, -1, 1, omega, omega^2).
All arithmetic is exact in Q[omega]/(omega^2+omega+1). The diagonal twist
matrix T obeys (ST)^3=S^2, and the Gauss sum
sum_i d_i^2 theta_i is six, giving the trivial chiral phase expected for a
quantum double.
Verlinde's formula applied to this independently constructed S matrix
reproduces the multiplicity-free fusion rules obtained earlier from the Hopf
coproduct and commuting-pair characters. In particular:
C*C = A+B+C
D*D = A+C+F+G+H
D*E = B+C+F+G+H
F*G = C+H
The checker passes seven aggregate gates with exit zero; fresh stdout matches the saved JSON after newline normalization.
Boundary
The modular pair records closed-link pairings and twists. It does not choose
trivalent fusion-space bases, microscopic intertwiners, F symbols, or
channel-wise R symbols. It therefore supplies an exact coherence
falsifier, not a proof that a particular microscopic F/R gauge satisfies
pentagon and hexagon. No uniqueness of braided category from modular data is
claimed.
Primary source boundary: Koornwinder--Schroers--Slingerland--Bais,
math/9904029, and Kitaev, quant-ph/9707021.