2265 — Eight Cyclic Branches Are Optimal for D(S3) Sector Dephasing
Sector: Kitaev (dephasing cost / branch calibration)
Claim
After admitting the one extra sector-separating endpoint port, a single cyclic
phase twirl dephases all eight D(S_3) sectors with exactly eight branches.
Eight is necessary because the sector eigenvalues require eight distinct
residues, and it is attained by
whose residues modulo eight are (0,1,2,3,6,7,4,5).
The earlier 19-branch witness remains valid but is not minimal. The sharp search must use the saturated integer lattice defined by the primitive central constraints; denominator clearing alone misses the optimum.
If branch zero is overweighted by eta and every other branch is changed by
-eta/7, every nontrivial Fourier mode has residual magnitude
Thus nonuniform branch weights immediately resurrect coherence.
The exact Fourier system also makes the uniform law unique. An ideal single-draw branch source therefore has Shannon entropy exactly three bits.
Scope
Eight is a branch-cardinality minimum for one cyclic generator. It is not a minimum pulse count, runtime, energy, randomness entropy, or fault-tolerance cost.
Durable verification
- Packet:
research/kitaev/s3-optimal-cyclic-dephasing-and-weight-errors.md - Checker:
research/kitaev/checkers/check_s3_two_flux_lie_control.py - Result:
research/kitaev/results/s3-two-flux-lie-control.json(schema v2) - Exact result: seventeen aggregate gates pass; fresh stdout matches saved JSON.
- Epistemic graph event:
ev-000000003135-df6b9161-8ecf-4b0f-97b4-fc36368a7f02