Beyond Loopwise Schur Margins: A Sharp Full-Gram Law for the Multiloop Return Margin

Local loop-by-loop stability margins can remain reassuring even when the global robustness of a multiloop system approaches zero. This paper identifies the information lost in Schur-reduced loop diagnostics and develops an exact full-Gram framework for recovering the corresponding global return margin. We establish two-sided elimination-tail identities linking local Schur complements to the input and output sensitivity Grams. The resulting subset margins have an exact perturbation-radius interpretation and satisfy sharp, dimension-dependent bounds relating local and global robustness. Explicit constructions demonstrate the optimality of these bounds and expose the limitations of conventional local diagnostics. We further develop a port-based tomography procedure that reconstructs the full Hermitian sensitivity Gram from a minimal set of scalar energy measurements, together with quantitative reconstruction-error bounds. A three-loop example demonstrates that phase information invisible to real-valued measurements and lower-order margin audits can alter global robustness. The framework also extends to stable linear time-invariant systems through frequency-domain sensitivity analysis. The results provide a precise connection between loop elimination, coherent sensitivity, measurement completeness, and multiloop robustness.