The Next-to-Highest Photon Threshold in a Bosonic Partial-Transpose Norm Conjecture

For the partial-transposed threshold projectors appearing in the bosonic PPT-assisted capacity framework of Barber and Pirandola, we prove an exact operator-norm identity for every number of modes and every pair of photon numbers when the threshold is one below the maximum possible matching number. An explicit two-term formula for the complete irreducible spectrum reduces the proof to coefficientwise positivity of a rational binomial-product polynomial; all nonaligned sectors have strictly smaller absolute eigenvalue. The result covers an infinite intermediate-threshold family, not the unrestricted threshold conjecture. Independent exact arithmetic and finite Fock-space checks accompany the analytical proof. Historical priority and external peer review remain to be established.