Eventual Partial-Transpose Norm Equality at Photon Threshold Two for Arbitrary Photon Numbers

We prove an exact partial-transpose operator-norm identity for the Barber–Pirandola correlation-threshold projector at threshold h=2 for every fixed pair of photon counts m,k≥2 and all sufficiently large mode counts s. The proof combines an all-mode one-sided sector inequality, a strictly positive second-order reciprocal-mode coefficient and an explicit constructive integer-polynomial cutoff. The result does not prove the full finite-block norm conjecture for arbitrary mode counts or thresholds. Novelty/priority and peer review are unverified; supplementary exact symbolic and rational-arithmetic verification programs accompany the manuscript.