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

For the bosonic threshold projectors studied by Barber and Pirandola, we prove the exact partial-transpose spectral norm identity at the balanced next-to-next-highest threshold: all integers s≥2 and m=k≥3 with h=m−2. The aligned sector reaches the norm; every other irreducible sector has strictly smaller absolute eigenvalue. An exact three-coefficient reduction and coefficientwise positive polynomial certificates prove the full unbounded family, supported by independent exact-rational spectral and symbolic tests. This result concerns a distinct threshold from our earlier h=min(m,k)−1 result, and neither proves the unrestricted norm conjecture nor changes the established asymptotic bosonic capacities. Historical priority and external peer review remain unconfirmed.