C073 - The thresholding miscount is corrected via P(up,up) = [(1-p)P(2) - p P(1)]/(1-p-q) and P(...
Verdict: partial (mismatch)
Location: Supp. Mat. S5, Eqs. (S7-S8)
Type / expected artifact: math / math
Claim: The thresholding miscount is corrected via P(up,up) = [(1-p)P(2) - p P(1)]/(1-p-q) and P(up,down)+P(down,up) = [(1-q)P(2) - q P(1)]/(1-p-q), where p (q) is the probability that 1 (2) bright ions are recorded as 2 (1).
Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial.
Source location(s): source/supp_content.tex:137-142 (Eqs. (S7-S8)).
Conclusion
Eq.(S7) P(uu)=[(1-p)P(2)-pP(1)]/(1-p-q) verified exactly by inverting the confusion matrix [[1-p,q],[p,1-q]]. Eq.(S8) as printed [(1-q)P(2)-qP(1)]/(1-p-q) does NOT match the inverse; correct form is [(1-q)P(1)-qP(2)]/(1-p-q) (P1/P2 swapped). Paper's S7+S8 violates probability conservation; the correct pair conserves it. Likely a typo in S8 -> partial (mismatch on second equation).
Verification details
Derivation excerpt: Let the true populations be $$T_1=P(\uparrow\downarrow)+P(\downarrow\uparrow)\ \ (\text{1 bright ion}),\qquad T_2=P(\uparrow\uparrow)\ \ (\text{2 bright ions}).$$ The supp. defines $p$ = probability a 1-bright event is recorded as 2-bright, and $q$ = probability a 2-bright event is recorded as 1-bright. (0-bright vs 1-bright misclassification is negligible.) The recorded probabilities are then $$P(2)=(1-q)\,T_2+p\,T_1,\qquad P(1)=(1-p)\,T_1+q\,T_2,$$ i.e. $$\begin{pmatrix}P(1)\\P(2)\end{pmatrix} =\underbrace{\begin{pmatrix}1-p & q\\ p & 1-q\end{pmatrix}}_{M} \begin{pmatrix}T_1\\T_2\end{pmatrix}.$$
Executable rerun: sympy_check.py exited 0 in 0.612s; log verification/C073/attempts/R002/sympy_check.log.
Output excerpt:
det M = 1-p-q: True
T1 (1-bright) = (P1*(q - 1) + P2*q)/(p + q - 1)
T2 (2-bright) = (P1*p + P2*(p - 1))/(p + q - 1)
S7 P(uu)==T2 (derived): True
S8 (as printed)==T1 (derived): False
T1 == (1-q)P1-qP2 form: True
derived T1+T2 == P1+P2: True
paper S7+S8 == P1+P2 : False