Manuscript I-B-VP-0@1 · I-B-VP-0-v1.tex · section · 8a9d0682

Ledger update and remaining questions

Ledger update and remaining questions · lines 1244–1347

{}1.1

Extension/state	Current status	Boundary
Extension/state	Current status	Boundary
Friedrichs, charged distinguished spectral ground state	Constructed and evaluated	Theorem thm:current; all $0 < α < 1$ , $0 \leq β < 1$ , on $r > 0$ .
Friedrichs, untwisted complex spectral ground state	Zero current	The covariance, Wick square, and stress tensor need not vanish.
Twisted non-Friedrichs $E_{0}, E_{λ}$	Open	No state or current evaluation is supplied here. Nonnegativity alone does not settle infrared finiteness.
Massless untwisted logarithmic $L_{L}$	Outside the nonnegative square-root prescription	Its transverse negative eigenvalue generates a negative longitudinal spectral band.
Mass-shifted logarithmic or negative- $λ$ families	Open	The positivity and threshold infrared qualifications of I-B-G/I-B-VP remain.
Thermal states, spinors, constrained gauge systems	Open	No such state or constrained apex domain is constructed.

The update to I-B-VP-v2 is restricted to the current-evaluation part of V4. A direct comparison with one current reference has also been supplied in Section sec:literature; a broader literature audit is still outside the result. The remaining scopes are unchanged:

Open: prescribed distributional extensions of the local observables across $W$ , including apex-supported counterterms. No such extension of the current is selected here.
Open: a complete global microlocal-spectrum analysis on the regular non-globally-hyperbolic host.
Open: the non-Friedrichs state and local-observable families, including massive logarithmic thresholds.
Evaluated here: the charged Friedrichs spectral ground-state current. A comprehensive current-literature audit remains a separate task.
Open: an apex curvature-coupling model selecting or modifying the extension.
Open: the corresponding massive-field companion construction and observable calculations.
Open: spinors and constrained gauge systems.
Open: thermal and other non-ground states.

Here Open means unresolved in this manuscript or the stated companion scope, not unknown throughout the literature.

at the apex. The current is a smooth one-form and therefore a distribution on $M$ . Its orthonormal component behaves as $r^{- 3}$ when the coefficient is nonzero. It does not define an ordinary locally integrable vector field across the completed transverse origin. This does not preclude distributional extensions; it means an extension prescription and its ambiguities have not been chosen. Regular-part conservation alone does not choose them.