An Exact Partial-Transpose Norm Identity at the Next-to-Next-Highest Bosonic Photon Threshold

We prove the partial-transpose operator-norm equality conjectured by Matthew Barber and Stefano Pirandola for the entire bosonic photon-correlation threshold subfamily h = min(m,k) - 2, for every number of modes s at least two and all photon numbers m and k at least three, including unequal photon numbers. The argument provides a three-term spectral reduction, universal exact positivity certificates at the two-mode boundary, and a polynomial mode-translation argument. Every nonaligned representation-sector eigenvalue satisfies a strict two-sided bound. A supplementary archive contains Python verification programs, exact integer-polynomial checks, regression tests against the original spectral formula, JSON verification records, and LaTeX source. The full arbitrary-threshold conjecture remains open.