OK, time for a list of known and potential improvements to the current plan. Might not save a whole turn yet, but it's getting close. For reference, the old plan can be viewed
here.
Amulet of life saving shenanigans
The first thing to note about the route is that the turn boundaries on Air and Astral are fixed in place. We can't do anything on Air without the turn boundary filling it with monsters, and thus in order to save a turn, the turn has to be saved entirely before the turn boundary in question ("2013:^" in the linked notes), or entirely afterwards.
However, we can fool this restriction somewhat. There's a glitch with amulets of life saving that means that certain commands can be caused to take no time if you die and are lifesaved during the command. (Specifically, this works only on commands that have no special handling for numeric prefixes.) This means that we can save one action before turn 2013 in the current route (2012 in a hypothetical 2014-turn route) via using the amulet of life saving action cancel glitch, but at the cost of one action after that turn (to put a new amulet of life saving on for Water). This is a good trade, because it lets us consolidate savings in one place rather than waste them to frame rules.
The action that we save before turn 2013 is actually one of the very first. On our second action of turn 2000, we level teleport down to the Dark One's lair. This action may well happen in natural form (see n720's post above, or my summary below), meaning that we can be vulnerable to HP death. One of our options for level teleportation is to invoke the Orb of Fate, which does a small amount of HP damage to us. If we're on critically low health at the time, we can die, lifesave, and cancel the action (meaning that we get six actions on turn 2000, which is probably a new record).
To pull this off, we also have to save an action somewhere on Air, Fire or Water. I'm currently unsure on how to do this, but there are plenty of potential suggestions (a promising one is to get pushed out of the water bubble we're in on the turn boundary after doing Air, thus saving us the first action of that turn; another possibility is to use air elemental polyform + the Eyes of the Overworld, with the disadvantage that it prevents using polymorph traps or level teleport traps on ourself earlier in the ascension run).
200 attempt teleport failure
People who have experimented with NetHack in debug mode generally know you can't teleport onto a trap. I knew that too, which is why I didn't factor it into my plans. As n720 points out, though, that isn't technically correct; if the game picks 200 sets of coordinates for a random teleport and they're all ineligible, the game starts looking at traps as options too.
This means that we can level teleport on the monster turn (something that isn't normally possible) via use of a quantum mechanic teleporting us onto a level teleportation trap (likewise, we can polymorph on the monster turn via the same mechanism). We can arrange for this to happen anywhere in Gehennom proper that isn't no-teleport, before we get the Amulet; the problem is that we can't have any MRtifacts in inventory at the time (the main problem artifacts being the Eyes of the Overworld and the Eye of the Aethiopica).
This doesn't directly save any of the actions in the plan I linked above. However, as n720 mentions, it allows us to rearrange actions on turns 2000 and 2001 to get the same result in natural-form-riding as it would in air elemental form:
2 actions lost to: the farmove actions needed to keep natural-form-riding happening at top speed.
1 action gained by: quantum mechanic teleport + 200 attempt teleport failure to enter the Sanctum; there's some subtlety here because monsters don't move after you enter a level on a monster turn, which n720's plan solves via doing the gallop farmove on that turn; if we combine this with the action saved to a "oLS then the timings don't work out correctly for that any more but there might nonetheless be some way around it via rearranging actions or saving some other action with a "oLS.
1 action gained by: falling off our steed onto the Amulet of Yendor at turn boundary, rather than having to step onto it manually.
So a wash, right? Actually no; air elementals are speed 48, a galloping unicorn is speed 53 (but is slower for general usage because you need to spend the last action of each turn moving, which is normally an inefficient use of time). We can spend three turn boundaries galloping this way, when with the old plan we'd only spend one. Perfect movement subpixels at turn 2000 are 11. 11 + 53 + 48 + 48 = 160 = 12*13 + 4, i.e. we have 13 actions total on turns 2000, 2001 and 2002 in the old plan. 11 + 53 + 53 + 53 = 170 = 12*14 + 2, i.e. we have 14 actions total on turns 2000, 2001 and 2002 in the new plan.
So the upshot of all this (besides making the TAS
way harder to make because the monster actions happen at some of the most awkward possible times that are theoretically possible to hit) is that we can gain one action via this particular string of coincidences, assuming that there's some fix for the problematic details.
Portal + level teleport to sequence break the Wizard's Tower
This was actually the first sequence break I discovered in NetHack, then I stopped using it in favour of enexto. It's a lot more stylish, and possibly easier to set up. But does it save time?
The original plan involved dropping the Amulet, then using a wand of teleport to teleport it out of the tower, but that's rather slow. There's an alternative, though; we can have a covetous monster steal it from us and then covetous-teleport out of the tower. (The only potential covetous monsters that will do this are doppelgangers of quest nemeses, with only a 5% chance of succeeding even if they use the attack in question, but we've never let the absurdity of something like that stop us before.)
If we enter the portal to wizard3 on our own action, then we can get engulfed and unengulfed on the following monster action, dropping us back on the same portal and sending us back to fakewiz1. The previous plan involved enextoing onto the portal (a monster-turn thing), but we could instead use punishment dragging to cover the short distance we need (note that we haven't been an air-E at any point before this turn, so we could set up a punishment ball before turn 2000 and just leave it in inventory, weighed up by the antigold). Note that we can't use jumping because that costs a whole turn.
This means that in both the old and new plans, the Amulet ends up on wizard3 the turn after we enter fakewiz1. The difference is where
we end up (wizard3 in the old plan, fakewiz1 in the new plan). The followup in the old plan is to enexto out of the tower onto a polytrap and turn into air-E form (we need to get out of p/t form). The corresponding followup in the new plan is to levport to wizard3, then on the monster turn have the covetous monster teleport over to us, and get killed by another monster, dropping the Amulet next to us. Unfortunately, even if the Amulet is dropped onto a polytrap (the optimal case from our point of view, as we want to become an air-E during this turn), we then probably have to spend an action picking it up, making this plan one action slower despite everything (sadly, we can't use the monster turn for help because wizard3 is no-teleport, and monsters with the Amulet never follow across levels).
It is probably possible to "oLS-cancel the action on which we pick up the Amulet (say, via autopickup of a cockatrice corpse without gloves), meaning that this option may be competitive if we need to find some other opportunity for "oLS-cancelling than the one on turn 2000. I'm very sad that I can't find a way to make it faster, though.
Another possibility is to somehow get the covetous monster in question to leave the tower on the monster turn during the first visit, after stealing the Amulet, so that it can die there. The only potential for this involves the 24 subpixel movement alternation glitch, and somehow levporting a nymph who has stolen the Eye of the Aethiopica to the level. Perhaps not impossible, but it'll depend heavily on where the turn boundaries are, maybe requiring "oLS-cancelling or the like to move them around.
Still another possibility involves riding for five turns rather than three. The problem is that riding for four turns is just outright cost-ineffective, and by turn 2004 (the fifth turn) we should be chain-quaffing "oLS.
Conclusion
Even if these strategies can somehow be made to work, we're still only saving three actions before turn 2013. To get a 2014-turn run, we need to save four. Thus, we still need to find extra savings somewhere, or else 2015 is still the theoretical limit. 2014 is not looking as insurmountable an obstacle as it did a few weeks ago, though.
Anyone got any more ideas?
The answer is half the difference between the bounds (10 and 30 in this example), meaning that by changing which numbers you put there, you can easily make it produce any answer you like. The mathematicians in this formula should see why it's a joke if they spend a few minutes trying to evaluate it, but I doubt many people will instantly work out what's going on there.