This work provides the analytical completion of a research series investigating the information-theoretic origins of quantum correlation kernels. While previous parts of this series (v1-v3) established extensive numerical evidence and geometric robustness for the correspondence between Information Causality (IC) and kernel positivity, this paper presents the first exact, constructive proof for all Fourier harmonics k 1. We introduce the Eigenvector Protocol, a physically valid N-bit Random Access Code (RAC) that utilizes the minimal eigenvector of the correlation matrix as measurement weights. Unlike previous approaches, this protocol is intrinsically sensitive to any negative harmonic, including higher-order terms (k 3) that were previously considered an open challenge in the literature. We prove that for any non-PSD kernel (characterized by a negative Fourier coefficient cₖ < 0), the IC-sum S = ₌₈₍² / |u|₁² strictly exceeds 1 for a sufficiently large protocol width N, while remaining 0 for all PSD kernels. This result closes the axiomatic gap, establishing Information Causality as the sole principle enforcing the full positive semi-definite (PSD) structure of quantum mechanics. Key Highlights: •Analytical Proof: Covers all harmonics k 1, solving the long-standing problem for k 3. •Constructive Protocol: Introduces the Eigenvector Protocol as a universal detector for non-PSD correlations. •Series Integration: Directly connects to the "TurboScan v6" numerical results and the geometric analysis presented in Part3.
Daniel Süß (Mon,) studied this question.