@awemany it is entirely possible that it's me not understanding something obvious, it happens all the time
Let me see If I'm able to explain what I meant more clearly.
I'm an attacker with an
X amount of hash rate, such amount will guarantee me to have a probability
p to "open" the sticky gate, i.e. mining AD+1 block before the rest of the network.
Let say that
p=0.2, that means that on average it took me 5 (
1/p) trials to get the first success, being able to open the gate.
For the sake of simplicity I'm going to assume that all the BU miners node is signaling EB=2/AD=5.
This how I see a failed attempt at opening the gate:
1 - the attacker will produce an excessive block (let say 3MB) on top of the current chaintip (CT)
2 - the rest on the network will start keep track of that "fork" but won't start building on top of
of it before another AD # of blocks will mined on top of it faster than what i mined on the main
fork.
3 - Unfortunately for the attacker the hash power on the main fork was able to produce AD+1 blocks
on top of the original chaintip (CT)
4 - The "honest" part of the network will stop monitoring the minority chain because the main
fork shown to have more PoW and being longer.
5 - In the process the attacker had mined 2 blocks on top of the initial excessive block. So this
mean that he wasted 12.5 * 3 bitcoin in this attack.
Repeat pints 1 to 5 four times, again for simplicity we suppose that every time the attacker waste 3 blocks.
The 5th time he was lucky enough to being able to mine AD blocks on top of the EB block and open the sticky gate.
In the first 4 trials he "spent" 12.5 * 3 btc on the 5th he spent 12.5 * (AD + 1) btc = 12.5 * 6 btc.
So it seems to me that it's not that the attacker is going to stop early to minimize the attack cost, it's just that once the main fork will be extended by AD+1 on top of CT before the attacker forked chain he has to restart again because his chain will be discarded by the honest majority.
edit: s/minority/majority/ in the last line