[[The One Ring]] is worded that you lose life, not pay life, so you should be able to still use at negative life totals.
#Can't activate The One Ring at negative life totals
late musk
distant copperBOT
Legendary Artifact
Indestructible
When The One Ring enters, if you cast it, you gain protection from everything until your next turn.
At the beginning of your upkeep, you lose 1 life for each burden counter on The One Ring.
: Put a burden counter on The One Ring, then draw a card for each burden counter on The One Ring.
late musk
Actually, finally noticed the [[Damping Matrix]] on the field. Disregard.