Prescribed geometry, bundle, and spectral state

Its Hilbert space is ${\mathcal{H}}_{\beta }={L^2}_{\beta }(N,\mathrm{d}V)$. The angular functions and radial Bessel orders are

The Friedrichs form is the closure of ${\int }_N|Du{|}^2\mathrm{d}V$ on compact smooth sections. The extension is an input, not a consequence of the regular metric CheungIBG,KayStuder.

and developing frame. With

and completion metric $\eta =\mathrm{d}{t}^2-\mathrm{d}{X}^2-\mathrm{d}{Y}^2-\mathrm{d}{z}^2$,

This is the equation-operator ledger of I-B-G and the ST sheets CheungIBG,CheungXAST,CheungXBST. An inverse of an action Hessian $-P$ is $-G_{\pm }$, not $G_{\pm }$. The current is defined directly from its Noether expression below, without importing a Hessian sign.