Spectrum, positivity, and scale

A transverse eigenvalue does not remain an $L^2(N)$ eigenvalue after adding the noncompact $z$-direction. It produces the branch

a continuous band containing negative spectrum. This distinction is essential in the instability argument.

In a finite chart, $H_{jk}$ has length dimension ${\mathrm{length}}^{-{\nu }_j-{\nu }_k}$. Thus $H_{0,-1}$ has dimension ${\mathrm{length}}^{-1/\alpha }$. Generic finite off-diagonal data introduce a length. The half-holonomy scale-invariant line family is a boundary relation and is not an exception represented by a finite nonzero off-diagonal $H$.