Three Formal Binary Settings Retain Faithful `D(S3)` Readout
Sector: Kitaev (non-Abelian instruments / sampling)
Artifacts: research/kitaev/s3-binary-hadamard-probe-instrument.md, its
checker, and research/kitaev/results/s3-binary-hadamard-probe-surface.json.
Claim
Refine the complex scalar probe surface into formal atomic binary settings
{Re twist, Im twist, S_A,...,S_H}.
Exhaustive search still gives minimum cardinality three. Exactly four minimum families remain, obtained from the earlier scalar minima by replacing full twist with its imaginary quadrature. A representative is
(Im twist, S_D, S_F).
The real twist quadrature is unnecessary. The imaginary quadrature alone
separates G/H; S_D and S_F separate the remaining labels.
For the formal Bernoulli completion, the binary plus probability is
(1+m)/2. The selected expectation signatures have minimum
distance 1/3, so their plus-probability signatures have minimum distance
1/6. Empirical plus-frequency error strictly below 1/12 preserves unique
nearest-signature classification.
Under independent, stationary, exactly calibrated Bernoulli repetitions, Hoeffding plus a union bound over the three settings gives
P(max_j |p_hat_j-p_j| >= 1/12) <= 6 exp(-n/72).
Thus ceiling(72 ln(6/alpha)) shots per setting suffice for failure
probability at most alpha. Eight exact gates pass, and fresh stdout matches
the saved JSON.
Boundary
Correction: the raw S_D,S_F effects are not established by a Hadamard-test
dilation. Hadamard tests return unitary matrix elements or normalized
traces, whereas mixed monodromy obeys tr M=6S_ab before internal-dimension
normalization. The declared Bernoulli effects are therefore a formal
completion pending a block encoding or independently derived interferometer.
The sampling bound remains conditional on its stated i.i.d. calibration
assumptions and is not a source-derived physical noise model.