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 non-compact 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 non-compact 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 transform ∆ ←→2 - ∆ of celestial conformal field theory, which identified as the boundary theory's version of the fundamental CPT-symmetry of regular four-dimensional quantum field theory in the bulk provides a means of resolving the googly problem regarding the perceived asymmetry problem in twistor-theoretic quantum gravity. 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 one mathematical fact: non-compact Haar measure gives a Plancherel axis, and compact Haar measure gives a Peter–Weyl gap.