Normalized Monodromy Restores a Typed Three-Setting `D(S3)` Instrument
Sector: Kitaev (non-Abelian instruments / controlled trace)
Artifacts: research/kitaev/s3-normalized-monodromy-controlled-trace-probes.md,
its checker, and
research/kitaev/results/s3-normalized-monodromy-probe-surface.json.
Claim
The apparatus correction in ledger 2196 has a constructive replacement. A
controlled-unitary trace test does not measure raw S_ab; on a maximally
mixed internal tensor-product space it measures
mu_ab = tr(M_ab)/(d_a d_b) = 6 S_ab/(d_a d_b).
All normalized monodromy expectations lie in [-1,1] and therefore define
direct Hadamard-test binary probabilities (1+mu_ab)/2 under the stated
controlled-unitary and state-preparation capabilities.
Over the corrected atomic surface
{Re twist, Im twist, mu_A,...,mu_H},
three settings remain necessary and sufficient. Exactly four minimum
families exist, with representative (Im twist,mu_D,mu_F). Its expectation
signatures for A,...,H are
(0, 1, 1)
(0, -1, 1)
(0, 0, -1/2)
(0, 1/3, 0)
(0, -1/3, 0)
(0, 0, 1)
(sqrt(3)/2, 0, -1/2)
(-sqrt(3)/2, 0, -1/2).
The minimum expectation gap is 1/2, the binary probability gap is 1/4,
and empirical-frequency error <1/8 preserves unique classification. Under
independent stationary calibrated Bernoulli repetitions,
P(max_j |p_hat_j-p_j| >= 1/8) <= 6 exp(-n/32),
so ceiling(32 ln(6/alpha)) shots per setting suffice. Seven checker gates
pass and fresh stdout matches the saved JSON.
Boundary
The abstract dilation is typed conditional on coherent controlled monodromy or twist and maximally mixed preparation on the relevant internal space. No local fault-tolerant controlled-ribbon synthesis or sector-uniform internal state preparation is proved. The statistical bound remains conditional on its explicit i.i.d. calibration assumptions.