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Yup. Technically you take the damage and get healed for 2x of it (so you effectively gain life equal to damage taken), but False Promise triggers will only notice the overall change in life and treat it as a heal like you'd expect. You take the damage, then get triple healed, then get double unhealed, then get triple healed again, then take the damage again. Assuming you aren't too close to max HP, you should get effectively triple the normal heal from the spell. For instance, if you are at 500/1000 HP and take 100 pure damage during Borrowed Time and False Promise and nothing else during either, you will first register the 100 damage (which will be stored by False Promise), then be healed 200 by Borrowed Time (HP 500-700) then be healed 100 by False Promise (HP 700-800) then actually take the 100 damage (HP 800-700) then be unhealed back to 500 (HP 700-500) with that 200 "heal" saved. At the end of False Promise, you will first take double the saved heal (HP 500-900) then the 100 saved damage (900-800). In total, the 100 damage will have healed you 300 HP. False Promise can never hurt you, that's for sure. In the normal case where you have False Promise but not Borrowed Time or any other triggered heal, it will record damage you take, heal it back (so your HP stays "fixed" for the duration of False Promise), then deal it again at the end of the duration. Moreover, it will record any time your HP increases ("heals"), decrease it back to what it was before ("unheals," again to ensure your HP is fixed), then give you the heal again at the end of the duration, doubled.
So yes, you will always benefit from False Promise as long as you are not already at max HP, in which case the doubled healing will of course be meaningless.
If Borrowed Time outlasts False Promise, the final damage will be healed by Borrowed Time just like any other damage. Keep in mind that the damage is done instance-by-instance, so assuming no single instance of damage exceeds Abaddon's max HP (in which case it should have killed him when it was first dealt, unless his max HP has decreased since then due to e.g. Armlet of Mordiggion deactivation or an enemy Undying casting Decay), he should always survive it and heal.
If you are affected by both Ice Blast and False Promise, you will get most of the effects of both, including purging debuffs (except Ice Blast and some others) on False Promise's cast, permanent invisibility, HP freeze, and magical damage over time. However, the damage over time will be registered and delayed by False Promise.
Ice Blast will still shatter you if you take damage while below the threshold under the effect of False Promise.
If False Promise ends before Ice Blast ends, and the net result of all the instances of damage and all the doubled heals is that your HP would go up, Ice Blast will prevent it from rising (and instead it will remain constant).
If False Promise ends before Ice Blast and the net result is that your HP would go down, nothing special happens, unless before one of the stored damage instances your HP goes below the threshold, in which case the next instance will shatter you.
If you would be healed while under the effect of both spells and the Ice Blast check occurs next, you won't get the heal. If you would be healed while under the effect of both spells and the False Promise check occurs next, False Promise will register the heal and prevent it from happening immediately, so you will get double that heal after the end of False Promise's duration, as long as Ice Blast has ended by then. Ice Blast checks HP every 0.01 seconds, while False Promise only checks every 0.02, so 75% of the time, the Ice Blast check will occur next and you will not get the heal. (Math is explained here.)
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The easiest way to convince yourself of this is to consider it as two independent events. If the False Promise timer starts at t=0, then if the heal occurs somewhere between 2n s and (2n+1) s, it cannot be registered by False Promise, because an Ice Blast check must come first. If it is between (2n+1) s and 2(n+1) s, you have an even probability of the heal coming before the Ice Blast check or after it.
The level makes no difference. If you time the heal ignorant of either timer (which is always the case, because the timers are so fast), you have a 25% probability of receiving the heal (doubled) at the end of False Promise, and a 75% probability of not receiving it.
In order to shatter due to Ice Blast, your HP needs to be below the threshold before some instance of damage. So for instance, if I have 101/1000 HP and am hit by a level 1 Ice Blast (10% threshold), then take 2 damage, I will not shatter, because my HP was above the threshold before that 2 damage was dealt. However, now my HP will be 99/1000, which is below the threshold, so if I take any more damage, I will shatter.
But in the case of Ice Blast + False Promise, False Promise generally won't let your HP drop in the first place (from damage anyway; HP removal can still reduce it). So if you are above the threshold, you are likely safe from shattering for the remaining duration of False Promise.
However, if you are already below the threshold, or if you cross the threshold due to HP removal, then you will still shatter at the next instance of damage, and False Promise cannot prevent that shattering (only Shallow Grave can).
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