Explicit Electric Intertwiners Produce a Microscopic `D(S3)` `F/R` Fragment
Sector: Kitaev (non-Abelian quantum doubles / microscopic coherence)
Artifacts: research/kitaev/s3-electric-FR-coherence-fragment.md,
research/kitaev/checkers/check_s3_electric_recoupling.py, and
research/kitaev/results/s3-electric-recoupling.json.
Claim
The pure-electric sectors A,B,C of D(S3) form Rep(S3). Freeze the real
two-dimensional standard representation C, solve all Clebsch--Gordan
intertwiner equations exactly, normalize the embeddings, and fix their signs
by a declared first-nonzero-entry convention. The decomposition
C*C = A+B+C
is orthogonal and complete. On the three-dimensional multiplicity space for
C*C*C -> C, overlap of the left- and right-associated microscopic
embeddings gives
F_CCC^C = [[ 1/2, 1/2, 1/sqrt(2)],
[-1/2, -1/2, 1/sqrt(2)],
[1/sqrt(2), -1/sqrt(2), 0]].
This matrix is exactly orthogonal. It is derived from representation matrices and intertwiners, not reconstructed from modular data.
Electric braiding is tensor-factor exchange. Its channel eigenvalues are
(R_A^CC, R_B^CC, R_C^CC) = (1,-1,1).
Writing B12=diag(1,-1,1) and B23=F B12 F^T, exact symbolic calculation
gives
B12 B23 B12 = B23 B12 B23,
B12^2 = B23^2 = 1.
The checker passes eight aggregate gates with exit zero, and fresh stdout matches the saved JSON after newline normalization.
Boundary
This is a microscopic coherence fragment for the electric subcategory, not
the full D(S3) category. It does not compute flux/dyon intertwiners,
mixed-sector F symbols, non-symmetric channel R symbols, or exhaustively
verify the full pentagon and hexagon families. Strict associativity of the
ambient vector-space tensor product is not substituted for those
simple-channel audits.
Carrier geometry supplies fusion-tree association and exchange paths; the quantum coefficient lens supplies representations, normalized intertwiners, inner products, recoupling coefficients, and channel braid actions.