Sharp dimension-independent Schatten bounds for PPT-protected coherence

How much coherence can remain when its two protecting diagonal blocks have positive partial transpose? This article gives a sharp bound without fixing the local Hilbert-space dimensions. Let H be a positive two-by-two operator block matrix with diagonal blocks A and B and off-diagonal block X. Each diagonal block is bipartite and has positive partial transpose internally. The coherence X may be rectangular, complex and non-Hermitian. Define two crossed arrangements by \(\mathcal R(X)_{ik,j\ell}=X_{ij,k\ell}\) and \(\mathcal G(X)_{i\ell,kj}=X_{ij,k\ell}\). Then \[\max\{\|\mathcal R(X)\|_{4/3},\|\mathcal G(X)\|_{4/3}\}\le\sqrt{\operatorname{Tr}A\,\operatorname{Tr}B}.\] The traces and Schatten norms are unnormalized. The constant one is optimal, and every smaller Schatten exponent fails uniformly in dimension. Internal PPT protection of the diagonal blocks is distinct from requiring the entire flagged block matrix to be PPT. Why the endpoint is sharp. Marginal spectral truncation gives a dimension-independent singular-value tail of order \(n^{-3/4}\). Closure under tensor powers removes the prefactor and returns the strong \(4/3\) endpoint. A Fourier shield family already present in the private-state literature supplies matching equality and the obstruction below the endpoint; the new contribution is the universal upper bound and its endpoint identification. Infinite-dimensional use. The paper extends the result to trace-class guards on separable Hilbert spaces, specifies the partial-transpose and local-filter domains, and states the joint weighted-trace conditions for unbounded filters. Trace-norm conversion and partial trace incur additional explicit costs. The integrated 10-page article includes two figure groups, complete proofs and references.