2249 — Two D(S3) Flux Ports Give Projective but Not Full Block-Unitary Control
Sector: Kitaev (dynamical Lie control)
Claim and correction
Hermitian gauge quadratures together with one transposition- and one three-cycle-flux projector generate a real Lie algebra of dimension
Full associative generation therefore does not imply full blockwise
U(d_a) control. Three independent central phase combinations are absent;
the available trace-signature rank is five rather than eight.
The derived commutator algebra is nevertheless complete projectively:
with block ranks (0,0,3,8,8,3,3,3). Hence every nontrivial block has full
special-unitary control. This is sufficient for conjugation twirls because
scalar phases cancel, but not for arbitrary coherent relative sector phases.
Scope
This exact finite Lie theorem does not derive pulse locality, amplitudes, time-ordering synthesis, leakage suppression, or fault tolerance from the Hamiltonian.
Durable verification
- Packet:
research/kitaev/s3-two-flux-projective-lie-controllability.md - Checker:
research/kitaev/checkers/check_s3_two_flux_lie_control.py - Result:
research/kitaev/results/s3-two-flux-lie-control.json - Invocation:
uv run --with sympy --with numpy python -u research/kitaev/checkers/check_s3_two_flux_lie_control.py - Exact result: eight gates pass; fresh stdout matches saved JSON.
- Falsifier residual: full block-unitary dimension deficit
36-33=3. - Epistemic graph event:
ev-000000003116-cb1a348a-018f-447f-8c11-77f945d7608f