Sharp Schatten constants for the arithmetic symmetric modulus: dimension two and rank-two strata

For the arithmetic symmetric modulus S(A)=(|A|+|A*|)/2, we determine the exact sharp Schatten constant in dimension two for every number of summands m>=2 and every finite interior exponent 1 2. The p=infinity endpoint is prior work and is used only as background. Two versions of record were unavailable during preparation, so no worldwide-priority claim is made. Status: Public Beta. Internal proof, bounded novelty, complete-manuscript and full-page artifact checks passed within their stated scopes. External mathematical review and formal peer review have not been completed. The archive contains the frozen manuscript, accepted TeX/PDF, twelve regression and destructive-control programs, reproduction instructions, artifact report and checksums. The separately downloadable PDF is identical to the accepted PDF inside the archive.