Abstract

This paper isolates a common spectral grammar behind three Millennium Problems which, in their classical formulations, appear to belong to different worlds. Each problem is associated with a shadow-symmetric spectral datum: a Hilbert space, an involution exchanging two spectral half-planes, and a fixed self-dual interface. For the Riemann zeta-function the interface is the critical line Re(s) = 1/2; for an elliptic L-function the interface is the central point s = 1; for Yang-Mills the interface separates the vacuum from the positive-mass spectrum. Given such a datum, exactly three primitive spectral questions arise: whether all spectral support lies on the interface (location); what multiplicity the interface carries at a distinguished point (multiplicity); and whether the positive spectrum is separated from zero (exclusion). In the Shadow framework, the Riemann Hypothesis, Birch-Swinnerton-Dyer, and the Yang-Mills mass gap are the arithmetic, elliptic, and gauge-theoretic realizations of these three questions. The paper proves the abstract spectral trichotomy, identifies each classical problem with its spectral role, and maps the common Haar positivity mechanism underlying all three. The purpose is not to compress the dedicated proofs into a single argument, but to explain why the three problems belong to one spectral family: RH fixes location, BSD computes multiplicity, and Yang-Mills proves exclusion.