  This study examines geodesics and plurisubharmonic envelopes within the
Cegrell classes on bounded hyperconvex domains in $\mathbb{C}^n$. We establish
that solutions possessing comparable singularities to the complex
Monge-Amp\`ere equation are identical, affirmatively addressing a longstanding
open question raised by Cegrell. This achievement furnishes the most general
form of the Bedford-Taylor comparison principle within the Cegrell classes.
Building on this foundational result, we explore plurisubharmonic geodesics,
broadening the criteria for geodesic connectivity among plurisubharmonic
functions with connectable boundary values. Our investigation also delves into
the notion of rooftop envelopes, revealing that the rooftop equality condition
and the idempotency conjecture are valid under substantially weaker conditions
than previously established, a finding made possible by our proven uniqueness
result. The paper concludes by discussing the core open problems within the
Cegrell classes related to the complex Monge-Amp\`ere equation.
