Hermitian Gram Calculus: Phase-Faithful Elimination, Transport, and Composition

Complex response families contain more information than can be recovered from norms, diagonal data, or real-valued reductions alone. Their relative phases and mixed correlations may change under elimination, transport, and changes of realization, so a common language is needed to determine what is retained and what is lost. This work develops a Hermitian Gram calculus that organizes these operations within a single geometric framework. It treats phase-sensitive reduction, comparison between realizations, and composition of response data while keeping intrinsic Hermitian information distinct from additional application-dependent structure. The framework connects local variations with history-dependent correlations, scale-free phase transport, and multipoint geometric information. It provides quantitative stability criteria, exact compatibility relations, and sharp boundaries for the regular regime. The main conclusion is that the full Hermitian Gram structure is the natural level at which complex responses can be reduced, transported, and compared without discarding information needed to distinguish systems that appear identical under scalar or purely real diagnostics.