C067 - Turning every odd drive (C) into a pi-pulse acts as a spin-echo that coherently reverses...
Verdict: verified
Location: Supp. Mat. S4
Type / expected artifact: math / math
Claim: Turning every odd drive (C) into a pi-pulse acts as a spin-echo that coherently reverses the differential-phase error, returning the population to |Psi^-> at the end of each even cycle.
Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain verified -> verified.
Source location(s): source/supp_content.tex:127 (Supp. Mat. S4).
Conclusion
Spin-echo cancellation confirmed analytically and numerically. Per-drive-(A) error E(phi)=diag(exp(-i phi/2),exp(+i phi/2)) on {|up,dn>,|dn,up>}; collective drive-(C) pi-pulse X=sx(x)sx swaps them. Echo identity X E(phi) X = E(-phi) holds on the ground-state subspace (verified phi in {0.3,1.0,2.137,3.0}). Over an odd/even cycle pair U=E(phi) X E(phi)=X E(-phi) E(phi)=X, so phase cancels exactly and X|Psi->=-|Psi->: system returns to |Psi-> with P(|Psi->)=1.000000, leakage ~1e-33 independent of phi. Without echo, two plain A drives leak ~0.71 to |Psi+> at phi=2.137. Clean mathematical fact -> verified.
Verification details
Derivation excerpt: From C066, one application of drive (A) with the differential Stark error imprints, in the $\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\}$ basis, the unitary $$E(\phi) = \mathrm{diag}\big(e^{-i\phi/2},\,e^{+i\phi/2}\big), \qquad \phi = 2\epsilon\Delta t,$$ which couples $\ket{\Psi^-}\!\leftrightarrow\!\ket{\Psi^+}$ via $E(\phi)\ket{\Psi^-}=\cos\tfrac{\phi}{2}\ket{\Psi^-}-i\sin\tfrac{\phi}{2}\ket{\Psi^+}$.
Executable rerun: run.py exited 0 in 0.53s; log verification/C067/attempts/R002/run.log.
Output excerpt:
X|up,dn>=|dn,up>? True
X|up,up>=|dn,dn>? True
-- echo identity X E(phi) X = E(-phi) on ground-state subspace --
phi=0.300: X E X == E(-phi) on P_g? True
phi=1.000: X E X == E(-phi) on P_g? True
phi=2.137: X E X == E(-phi) on P_g? True
phi=3.000: X E X == E(-phi) on P_g? True
-- one odd/even cycle pair on |Psi-> --
phi=0.300: P(|Psi->)=1.000000 leak P(|Psi+>)=3.85e-34
phi=1.000: P(|Psi->)=1.000000 leak P(|Psi+>)=0.00e+00