Hi
Previously (in discord), i made topic about Vulthoo buildtime (bt should be reduced). In the process i went deeper into vulthoo stats and discovered that they are far from perfect.
So i decided to really understand what is going on here and what Vulthoo stats should be if we want it to be "T2.5" gunship, using math & logic to get the values. (I dont really want to do this already but i promised to Rhaelya so rip me )
Key points:

I assume that Vulthoo should be T2.5 gunship in all the stats (assuming mass cost is constant 500)

T2.5 gunship here is the gunship between Stinger and Broadword.
Why Broadswoard? It has no other abilities (Wailer's stealth, Restorer's AA), as well as Vulthoo
Why Stinger? Its the only t2 gunship which shoots as Vulthoo (no aoe, no alphashot)
I will mention some other gunships still tho 
I will split the post into several chapters to make the brain load less. Each chapter will have some thought about some particular param and what it should be.
Prologue, speed & range
The data:
As you see, currently vulthoo is slower than needed and has more range than needed.
Someone will say "but stinger is the fastest t2 gunship"
Ok, Specter & Renegade have 12 speed, lets calculate "average t2 gunship speed" then: (12 + 12 + 13.5) / 3 = 12.5.
Even with this approach, T2.5 will be (12.5 + 10) / 2 = 11.25
About range, Renegade has the same range as stinger, only specter has 20, which will lead to 23.16 range
So:
Vulthoo:
 speed should be buffed to 11.75 (11.25 if we look not only on stinger) (11.5 average between both suggestions)
 range should be nerfed to 23.5 (23.16 if we look not only on stinger) (23.33 average between both suggestions)
Interlude, about raw power
As soon as i will suggest any change, some people will appear to claim that "but concentraded power!"; Fair enough
To key points here:
 Concentraded power factor vs flaks: hard to calculate "middle value". I mean its obvious that vulthoo is better than stinger vs flaks, but what stats exactly should be? Size of the gunship? Range? Hp? This is too complicated to make any suggestions, so i completely skip this case because i have no idea how to analyze it properly so it makes any sense.
 Concentraded power factor vs anything else: this one can actually be calculated!
Most AA in the game has no aoe (Any air aa: inties, swiftwinds, asfs, t2 bombers, restorers, etc) or has small enough aoe to still completely negate its effect with proper micro (nonsera cruisers, sams, t3 maa) (note: use formation move and some move orders for ez negating aoe of any of these)
So, lets define raw_power as the amount of damage the group of N gunships can do before some ideal target with infinite HP and some singletarget AA kill them. And N can be calculated for each resource (mass, energy, buildpower) individually (how many gunships i can build for 5k mass / 50k energy / 50k buildtime, etc)
The formula is:
If we want to see only "real" game values (because we cannot send 3.41 gunships, we can send only 3), we can just floor the values:
For any people wondering why this formula is like this, i will just show this screen and i think this is enough.
Formula is correct according to tests, if you have any doubts free to write below
Chapter 1, DPS and HP per mass
The data:
Raw power, under 3000 mass (left is smoothed, right is "real"):
Raw power, under 20000 mass:
As you see, currently Vulthoo has lower hp/dps values compared to what it should be. In addition to that, it also has lower raw_power per mass (even if you count "concentrated power" effect, its worse)
Yes, in current game balance t2 gunships have better raw_power per mass than t3 at the cost of being bad vs flaks and buildtime difference (about thatlater).
Lets try to fix dps/mass according to the table:
dps: 100 > 111
hp: 1800 > 1911
The data:
As you see, now it looks more what is should be. On large scale it performs as t2.5 indeed. Under 1500 mass (3 gunships) spended it is very close to stinger by raw strength due to "high mass concentration" factor, and after that cloud of stingers starts to be stronger.
I believe these are right numbers to be. If you worried about "but i will be sniped before even build first flak, i understand your concern, but dont worry, we can balance this with e cost for example. Vulthoo is still much worse vs flaks than broadsword and it should be used on t3 because sera has no alternatives, so being too weak just is a bad idea.
So:
dps: 100 > 111
hp: 1800 > 1911
Chapter 2, buildtime
The data, [current vulthoo]
From the data, we can see two things:
 t3 gunships are far better in raw power per buildtime compared to t2. In addition to that, t3 fac has 3x more buildtime than t2, so you can multiply broadsword score by 3 if you want.
 Vulthoo buildtime is just stupidly bad currently. It supposed to be better than stinger in (raw_stength / buildtime) because of "T2.5", instead he nerfed below that.
I will draw suggested charts for current Vulthoo and for new Vulthoo. New Vulthoo will have all the stats i suggested above so we need to account for them while suggesting other stats.
What builtime current Vulthoo should have? That is a really tricky question!
Here is the plot, "how much builtime vulthoo should have to be T2.5 in raw_power_per_buildtime terms":
As you see, the answer is really depends on what amount of buildtime (BT) you use. (With current stats, no constant value of buildtime will make raw_power_per_buildtime always exactly between Broadsword and Stinger at any point of the game)
The max value (limit to the infinity, very large armies) of the function is ~ 2437.75 buildtime
The min value (limit to 0, tiny armies) is ~ 1347 buildtime
for BT = 3300: 2026
for BT = 10000: 2247
for BT = 30000: 2364
So, the actual value here should be taken from the point (how many BuilTime will be invested into Vulthoo in the average game)
I think it is not more than 3050k, so 23642400 should be good enough for current Vulthoo.
Here are the charts for raw power with 2370 buildtime vulthoo:
Now, lets go to more instesting part and check out or new Vulthoo (111dps, 1911hp):
The same situation with buildtime, it depends:
But here values are starts from 1587.370786 and going to 2646
T2.5 BT for 30k Buildpower invested is ~ 2575
T2.5 BT for 50k Buildpower invested is ~ 2600
Excel table suggested value which has no idea about "concentrated mass" is 2630, which is still in range
Ok, lets check out 2600:
Looks good
So
new Vulthoo:
dps > 111
hp > 1911
buildtime > 2600
Buildtime can be a bit more if you wanna balance it for ultralategame, but nothing more than 2646 make any sense.
Can also be a lot less if you wanna balance it for more early game usage instead.
Chapter 3, energy cost
The data, [current vulthoo]
Intresting! Finally, green numbers!
Now let me explain what is going on here:
 t3 energy is much cheaper than t2 energy. For each 100 mass invested into t2 pgen, you get 41.6 energy income.
For each 100 mass invested into t3 pgen, you get 77.16 energy income
So basically t3 energy is 1.85x cheaper than t2 energy  If we look at vulthoo as pure t2 gunship which is produced on t2 pgens, then current E cost makes sense: 1.85x multiplier nullifies broadsword's high E cost, so current E cost looks fine at 99% of what dps/mass per energy should be.
 if we forget about 1.85x multiplier and look only on E cost, basically producing all gunships at t3 pgens, then vulthoo has 113% dps / energy and 131% hp / energy it should have
 if we imagine some mythic T2.5 Pgen which is average between T2 and T3, then vulthoo has 106% of correct dps / energy and 119% of correct hp / energy
Since this whole topic is about making T2.5 gunship, i assume 4) as right point
T2.5 gunship should cost more energy than current Vulthoo costs
for current Vulthoo, its about 11200e.
But we're talking here not about current fraud t2.5, we're talking about real T2.5
So, what energy cost should have new Vulthoo with stats:
111dps
1911hp
2600bt
?
Its about 12200e (22% more than current) to be balanced for imaginary "T2.5 pgen".
If you want to balance for T3, its 13200
If you want to balance for T2, its 10700
These numbers are all from only dps and hp / energy stats, what about raw_power?
[current Vulthoo picture]
Here are 3 charts. Basically, all what adjusting does is shifts broadword ratio to stinger and vulthoo. adjusted_energy_for_t2 makes discount for broadsword cuz t3 energy is cheaper per mass.
Adjusted_energy_for_t2.5 makes also discount for vulthoo & stinger, imagining they are builded on t2.5 pgen (trying to balance both for t2 and t3 at the same time).
As you see, currently vulthoo is indeed considered t2 gunship from energy POV. It has not enough E cost to be t2.5, which can be partially responsible for its current horrible stats.
[new Vulthoo picture]
Yep. Now, spamming Vulthoo at t2 pgens will not give you more raw_power / [adjusted_energy] than spamming broadsword on t3, but using t3 pgens you still will have more, as it should be for t2.5 gunship.
So
e cost: 10000 > 12200
Conclusion
Current Vulthoo is not even near t2.5 gunship level.
Right T2.5 gunships stats are:
Speed > 11.75 (Or 11.25/11.5, see prologue)
Range > 23.5 (Or 23.16/23.33, see prologue)
Hp > 1911 (chapter 1)
Dps > 111 (chapter 1)
Buildtime > 2600 (or from 1587 to 2646, see chapter 2)
E cost > 12200 (chapter 3) (also see post below)
These are correct stats for T2.5 500mass gunship
From all the above only E cost is somewhat arguable, because the result is really depends on "do i adjust for t2/t3 pgens factor and if yes, how?". (Depending on how you adjust for unit tier, i got up to ~15300 e cost as middle one)
Make hp/dps/speed less and now its not t2.5 really but some strange abomination.