From local spectra to predictive response: exact information-sufficiency boundaries for integrated matrix systems

Description This record contains the complete manuscript, Supplementary Material, source files, and reproducibility package for the study “From local spectra to predictive response: exact information-sufficiency boundaries for integrated matrix systems.” The work establishes exact information-sufficiency boundaries for integrated positive matrix-valued physical responses. For a family of positive semidefinite matrices A(t) with nonnegative weight w(t), the integrated operator Λ = ∫ w(t)A(t)dt is shown to be the exact lossless matrix summary for all integrated quadratic readouts x*Λx. Complete local eigenvalue information is nevertheless not sufficient in general: even isospectral, pairwise commuting slice families can produce different integrated responses, with a sharp factor-d ambiguity in the rank-one case. The analysis further shows that pairwise projector-overlap magnitudes can remain insufficient in complex channel geometry because they can omit gauge-invariant cycle-phase information. For finite rank-one ensembles, the phase-aware weighted complex Gram matrix recovers the nonzero spectrum of the integrated operator, while a separate global-unitary embedding ambiguity explains why fixed-physical-basis directional responses require additional projector or factor information in a common physical basis. The work also derives the exact source-subspace compression PΛP and query-dependent computational consequences. The accompanying Supplementary Material provides complete technical proofs, equality conditions, sharp finite-ensemble constructions, the top-r Ky Fan extension, the cycle-phase obstruction, weighted-Gram sufficiency, embedding limitations, and detailed scope qualifications. All analytical results are established independently of numerical computation; the computational material is used only for regression tests, finite-precision verification, stress testing, figure generation, and reproducibility. The full reproducibility package includes the main manuscript, Supplementary Material, LaTeX sources, deterministic replay code, numerical outputs, generated data, figures, and audit files needed to reproduce the reported validation results. Files in this record: Main manuscript PDF and LaTeX source Supplementary Material PDF and LaTeX source Full reproducibility package ZIP containing code, numerical outputs, data, figures, and audit materials This archive corresponds to the complete V4 research release.