One Twirled Flux Reference Separates All Pure-Electric `D(S3)` Charges
Sector: Kitaev (non-Abelian instruments / reference twirling)
Artifacts: research/kitaev/s3-electric-one-setting-reference-twirl.md,
its checker, and research/kitaev/results/s3-electric-reference-twirl.json.
Claim
Uniformly conjugating a fixed transposition flux by all elements of S3
produces each of the three transpositions exactly twice. A fixed three-cycle
similarly produces each cycle exactly three times. This supplies uniform
conjugacy-class reference mixtures by classical random gauge conjugation.
On the standard electric charge C, the averaged actions are
(1/3) sum_transpositions rho_C(t) = 0,
(1/2) sum_three-cycles rho_C(c) = -I/2.
They are scalar, so the unknown electric target's internal density matrix drops out. No target-sector maximally mixed preparation is required.
A single normalized-monodromy setting using the twirled D reference has
mu_D(A)=1, mu_D(B)=-1, mu_D(C)=0,
and binary plus probabilities 1,0,1/2. It therefore separates all three
pure-electric sectors. The expectation gap is one, probability gap 1/2,
and unique empirical-frequency radius <1/4. Under the explicit independent
stationary assumptions,
P(|p_hat-p| >= 1/4) <= 2 exp(-n/8),
so ceiling(8 ln(2/alpha)) shots suffice. Seven exact gates pass and fresh
stdout matches the saved JSON.
Boundary
The result requires a constructible D reference, uniform random group
conjugation, and controlled monodromy. It does not prove local fault
tolerance or extend the state-independence result to flux/dyon targets. It
is a one-setting theorem only on A,B,C, not on all eight sectors.