[1] @Mengerian also made some plots that illustrate the mining gap in the previous Gold thread on BCT and possibly has insight into what happens as R/T --> 0.
Hi Peter! I'll try my best...
I think the issue with R --> 0 stems from the concept of "neutral profit" in your formulation. Because "neutral profit" is based on the miner's profit with on empty block, this becomes nonsensical when the block reward is zero, and thus the analysis breaks down.
I think the solution to this is to base the formulation on "zero profit" instead of "neutral profit". Take your equation (5) and set profit to 0. solve for M, yielding:
M=ηHTe^(τ/T) - R
This is the zero profit curve. When R = ηHT (cost = block reward revenue), it resolves into the same equation as your "neutral profit". Currently when block reward is the majority of revenue, we can expect R > ηHT. in this case, the curve will start below 0, so the profit will be positive for empty blocks, up until the lines cross at some large Q.
But when fees predominate in future (ie R << M), we should expect R < ηHT. In this case this "cost" curve will start above 0, so the "demand" curve (which starts at 0) will be below it, meaning that empty blocks are unprofitable. This is the "mining gap" situation.
Now in this formulation, equation (8) will become:
p_supply = ηH dτ/dQ e^(τ(Q)/T)
(ie, replace R/T with ηH)
Similarly equation (10) can be modified by replacing R/T with ηH. This makes sense intuitively, since the value ηH is simply the hash cost rate, the cost of maintaining the network hash rate per unit of time.
Anyway, I hope that all makes sense. I'll try to go through the paper more thoroughly when I have time.
By the way, this approach removes the assumption that block reward is non-zero, but assumes some sort of equilibrium is achieved where the miners have incentive to include transactions in successive blocks so that fees will pay for hash rate H. I haven't really thought through the dynamics on Justus Ranvier's point about the lack of incentive to extend the blockchain with successive blocks when there is no block reward.