Haar Spectral Duality: Plancherel Continua and Peter–Weyl Discreta with Applications to the Riemann Hypothesis and the Yang–Mills Mass Gap

Haar measure has two spectral regimes determined entirely by the compactness of the underlying group. On a noncompact group the regular representation decomposes by Plancherel measure over a continuous unitary dual; on a compact group it decomposes by Peter–Weyl summation over finite-dimensional irreducibles. We prove this dichotomy in the forms needed for two applications in the Shadow Framework. For the noncompact case we show that the half-density Mellin transform on L²(ℝ⁺, dx) is unitary exactly on the line Re(s) = 1/2, and that the functional equation ξ(s) = ξ(1−s) of the completed zeta function is the restriction of Haar inversion self-duality to the arithmetic spectrum via Tate's thesis. Under Δ = 2s, the same inversion is the shadow symmetry of celestial CFT, identified as time reversal T. For the compact case we prove that Haar projection onto SU(N)-singlets is the orthogonal projection onto the physical gauge sector, and that the Sugawara conformal weight of the first adjoint-current excitation gives a mass gap M = 2N/(k+N)·Λ_QCD > 0 in the continuum celestial Yang–Mills construction. The Riemann and Yang–Mills mechanisms are the two complementary spectral faces of a single source: noncompact Haar measure gives a Plancherel axis, compact Haar measure gives a Peter–Weyl gap.