Three Scalar Probes Are Necessary and Sufficient for `D(S3)` Sector Readout
Sector: Kitaev (non-Abelian readout / probe minimality)
Artifacts: research/kitaev/s3-minimal-jointly-faithful-scalar-probes.md,
its checker, and research/kitaev/results/s3-minimal-scalar-probe-family.json.
Claim
Freeze the candidate scalar probe surface to topological twist theta and
the eight reference-sector Hopf-link amplitudes S_A,...,S_H. Exhaustive
enumeration proves that no one- or two-probe family separates all eight
D(S3) anyon labels, while exactly four three-probe families do:
(theta,S_C,S_D), (theta,S_C,S_E),
(theta,S_D,S_F), (theta,S_E,S_F).
A representative minimum family is (theta,S_D,S_F). Its signatures for
A,B,C,D,E,F,G,H are respectively
(1, 1/2, 1/3)
(1, -1/2, 1/3)
(1, 0, -1/3)
(1, 1/2, 0)
(-1, -1/2, 0)
(1, 0, 2/3)
(omega, 0, -1/3)
(omega^2, 0, -1/3).
Twist resolves the centralizer-irrep kernel by separating D/E and all of
F/G/H. The two Hopf-link references separate the remaining electric/flux
collisions. Explicit lower-bound witnesses are: (theta,S_F) collides
A/B, (theta,S_D) collides C/F, and (S_D,S_F) collides G/H; the
checker additionally exhausts every other pair.
Seven aggregate gates pass with exit zero, and fresh stdout matches the saved JSON after newline normalization.
Physical and theorem boundary
The three probes require a framed self-twist constructor and prepared D and
F reference sectors supporting linked transport and fusion to scalar
records. Carrier geometry supplies framed/link topology and reference-port
placement; the quantum coefficient lens supplies sector preparation and
amplitudes.
Minimality is relative to this nine-probe surface and counts a complex scalar as one probe. It is not an absolute laboratory-resource theorem: full operator tomography, multi-outcome instruments, adaptive procedures, or a different quadrature cost model define different optimization problems.