  Information deformation and loss in jet clustering are one of the major
limitations for precisely measuring hadronic events at future $e^-e^+$
colliders. Because of their dominance in data, the measurements of such events
are crucial for advancing the precision frontier of Higgs and electroweak
physics in the next decades. We show that this difficulty can be well-addressed
by synergizing the event-level information into the data analysis, with the
techniques of deep neutral network. In relation to this, we introduce a
CMB-like observable scheme, where the event-level kinematics is encoded as
Fox-Wolfram (FW) moments at leading order and multi-spectra at higher orders.
Then we develop a series of jet-level (w/ and w/o the FW moments) and
event-level classifiers, and analyze their sensitivity performance
comparatively with two-jet and four-jet events. As an application, we analyze
measuring Higgs decay width at $e^-e^+$ colliders with the data of
5ab$^{-1}@$240GeV. The precision obtained is significantly better than the
baseline ones presented in documents. We expect this strategy to be applied to
many other hadronic-event measurements at future $e^-e^+$ colliders, and to
open a new angle for evaluating their physics capability.
