The Riemann Hypothesis and the Yang–Mills mass gap are usually discussed as separate problems, connected at most by the observation that both resist proof. The Haar spectral duality paper makes their connection precise: they are the two complementary spectral faces of a single source. The source is Haar measure, and the distinction between the two problems is the distinction between compact and noncompact groups.
Haar measure is the unique translation-invariant measure on a locally compact group. On a compact group, the total volume is finite and the representation theory is governed by the Peter–Weyl theorem: the regular representation decomposes as a discrete sum over finite-dimensional irreducibles. On a noncompact group of type I, the regular representation decomposes by the Plancherel measure over a continuous unitary dual. Compactness is the single structural property that determines which regime applies, and it determines the spectral answer completely.
The Noncompact Face: The Riemann Critical Line
The scale group (ℝ+, ×) is noncompact. Its Haar measure is dω/ω, and its Plancherel theory is the Mellin transform. The half-density Plancherel theorem establishes that the Mellin transform on L2(ℝ+, dx) is unitary exactly on the spectral line Re(s) = ½. This is a theorem of harmonic analysis that holds before any arithmetic enters. The functional equation ξ(s) = ξ(1−s) is then derived as Haar inversion restricted to the arithmetic spectrum, via Tate's 1950 thesis. Under Δ = 2s, Haar inversion is the shadow symmetry, which the Shadow Framework identifies as time reversal T. The admissible spectral support is the T-invariant locus Re(s) = ½.
The Compact Face: The Yang–Mills Gap
The gauge group SU(N) is compact. Its Haar measure is finite, the Peter–Weyl theorem applies, and the representation theory is discrete. Haar projection onto SU(N)-singlets is the orthogonal projection onto the physical gauge sector. In the boundary Kac–Moody algebra at level k, the Sugawara conformal weight of the first adjoint-current excitation is hadj = N/(k + N), strictly positive. The mass conversion M = 2hΛQCD gives the mass gap:
The Complementarity Theorem
The paper proves a complementarity theorem: any spectral problem governed by a Haar spectral datum lies in one of two regimes, determined entirely by compactness. If the governing group contains a noncompact scale factor, the spectrum is Plancherel-continuous and the unitary axis is Re(s) = ½. If the governing group is compact, the spectrum is Peter–Weyl-discrete and every non-trivial excitation has a positive spectral gap. The two cases are not analogous; they are complementary faces of one measure.
The paper also records a structural warning that follows from this theorem: the two faces cannot be confused. Compact discreteness cannot be imported into the arithmetic spectral problem to supply the missing Hilbert–Pólya step, because the relevant group ℝ+ is not compact. Any argument that attempts this import is formally invalid. The warning is stated precisely so that it cannot be evaded by rewording.
Where the Gap Remains
The Haar spectral duality paper establishes the admissible support for the Riemann zeros and the mass gap for Yang–Mills. It does not close the Hilbert–Pólya admissibility question. Whether the zeta zeros are spectral data of the self-adjoint dilation generator is the remaining open problem, stated as Definition 6.2 and Corollary 6.3 in the paper with complete precision. The distance between what is proved and what is claimed is zero; the distance between what is proved and a complete proof of the Riemann Hypothesis is exactly the Hilbert–Pólya step, no more and no less.
Haar measure has two spectral faces. The Riemann Hypothesis lives on the noncompact face: a continuous Plancherel axis whose admissible locus is Re(s) = ½. The Yang–Mills mass gap lives on the compact face: a discrete Peter–Weyl staircase bounded away from zero. One source. Two faces. The distinction is compactness, and compactness decides everything.