C001 - The singlet Bell state is defined as |Psi^-> = (|up,down> - |down,up>)/sqrt(2).
Verdict: verified
Location: Introduction
Type / expected artifact: math / math
Claim: The singlet Bell state is defined as |Psi^-> = (|up,down> - |down,up>)/sqrt(2).
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/main.tex:92 (Introduction).
Conclusion
Singlet |Psi^-> = (|up,down>-|down,up>)/sqrt(2). Verified normalized (norm=1.0), antisymmetric (SWAP|psi>=-|psi>), maximally entangled (reduced rho1=I/2, S=ln2=0.6931), total-spin-0 (Sx,Sy,Sz annihilate it). Clean math fact -> verified.
Verification details
Derivation excerpt: The claimed state is $$|\Psi^-\rangle = \tfrac{1}{\sqrt{2}}\left(|{\uparrow}{\downarrow}\rangle - |{\downarrow}{\uparrow}\rangle\right).$$
Executable rerun: sympy_check.py exited 0 in 1.19s; log verification/C001/attempts/R002/sympy_check.log.
Output excerpt:
norm = 0.9999999999999998
SWAP|psi> = -|psi> : True
reduced rho1 (qubit block) =
[[0.5 0. ]
[0. 0.5]]
reduced == I/2 : True
entanglement entropy S = 0.6931471805599454 (ln2 = 0.6931471805599453 )
Sx|psi> == 0 : True
Sy|psi> == 0 : True
Sz|psi> == 0 : True