A note on the Mellin spectral axis of the Weyl-symmetrized dilation generator: affine unitarity and the Berry–Keating programme

In the Berry-Keating operator approach to the Riemann zeros, the half-line Weyl-symmetrized dilation generator G = (1/2)(xp + px) plays a central role. We isolate a functional-analytic point often left implicit: the Mellin spectral axis Re(s) = 1/2 is fixed by the half-density constant in G. The logarithmic unitary sends G to ordinary momentum on L^2(R, du), giving generalized eigenfunctions x^(-1/2+iE) and spectral parameter s = 1/2 - iE, independently of the functional equation. We also address the measure-normalization loophole: relative to the canonical full-line affine (x, p) representation, translation unitarity selects Lebesgue measure, so the half-line representation inherits the half-density cocycle rather than choosing it freely. Finally, a minimal Krein-space doubling realizes the re-centered critical line as a kinematic spectral seam. The construction does not locate the zeta zeros; that requires a separate arithmetic positivity structure, such as a Hermite-Biehler/de Branges structure. The note separates the kinematic axis, lattice reflection, and RH-strength positivity strata of the H = xp programme.