the bleed

demurrage is a holding cost. every epoch, an idle balance is reduced by a fixed fraction of itself. anton reval sets that fraction at 2.00 percent and renounces the authority to change it. the reduction is not a fee paid to a treasury and it is not destroyed. it is swept into a redistribution pool and paid out to wallets that were active in the same epoch.

the per-epoch rule is multiplicative. a balance b that sits idle for one epoch becomes b times (1 minus r), where r is 0.02. after n idle epochs the balance is b times (1 minus r) to the power n. this is geometric decay, not linear. the absolute amount removed shrinks each epoch because it is always a fraction of a smaller balance, but the proportion removed is constant.

continuous compounding

the epoch model is the discrete face of a continuous process. if the same total fraction were applied smoothly across the epoch rather than at its boundary, the balance would follow b times e to the power of minus lambda t, where lambda is the continuous rate that satisfies e to the minus lambda equals (1 minus r) over one epoch. solving gives lambda equal to minus the natural log of (1 minus r), about 0.0202 per epoch. the discrete and continuous views agree at the epoch boundaries.

the half-life derivation

the half-life is the number of epochs after which an untouched balance has lost half its size. set (1 minus r) to the power h equal to one half and solve for h. h equals the natural log of (1 half) divided by the natural log of (1 minus r), which is the natural log of 2 divided by minus the natural log of (1 minus r). at r equal to 0.02 this is about 34.3 epochs, roughly 240 days at a 7 day epoch.