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5 
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I'll also point out that above 45 degrees elevation you trade range for altitude. If we graphed range vs elevation/altitude then we'd see a curve.
The British QF 5.25" Mk I page on Wiki has some useful data here.
Maximum range 22,010m
Ceiling 14,200m
Effective range 21,400m
Effective AA ceiling 11,00m
That was a more advanced weapon than some - and notice it's effective AA ceiling is half that of it's maximum range. That is quite a bit better than you report - but I don't know if there are other factors in play.
It would certainly seem there is evidence to improve the range of the HAA basket.
My preliminary assessment is that the effective range of HAA is 0.3 of the gun's range against surface targets.
However, ships who are in range with HAA (including the ship being targeted by the air attack) only have a chance to contribute to the HAA layer of defense (preliminary is that this is less than a coin flip, maybe p=0.3, need more data).
After all of that, there is some kind of significant penalty to the HAA effective factor that I haven't fully figured out, yet. I need more data but it seems to vary between about a quarter to a little more than half of sum of the HAA.
I suspect that having a mix of high HAA factor and low HAA factor ships (I.E. capital ships with large DP gun batteries with DD escorts carrying DP guns) is actually extremely punishing to the effective HAA factor.
However even with said FCS availabe gun can be defeated at longer ranges by simple non-linear maneuver because 40th-50th FC extrapolations are all linear. And if you take into account that all aircraft attack patterns are by their nature non-linear you will understand what HAA simply can't be anything but the fixed barrage at med-to-long range. And its almost zero kill probability and morale effect mostly.
But wait thats not all. Even morale effect was a thing only when aircrafts had to use large formations due to limited communications. After decent radios started to become available en-masse it also became possible to attack through multiple vectors making anything "fixed" only a waste of shells.
Once VT shells were available, the equation changed as they'd detonate in proximity, meaning they wouldn't quite be firing in fixed barrage. Rounds per kill dropped dramatically where VT shells were used.
https://www.history.navy.mil/research/library/online-reading-room/title-list-alphabetically/a/antiaircraft-action-summary.html#:~:text=Although%20the%20burden%20of%20ship,by%20naval%20and%20merchant%20ships.
Scroll down to find a table of kills by calibre of AA weapons and rounds per kill required..
The article you have linked clearly unable to distinguish "kill" and "claim". This is why 5" common in 1942 was supposedly more effective than 5" VT in 1945. Neither RPB number have anything to do with reality because nobody assessed actual kills by diving to the bottom of the ocean to check if it was true. Its just claims. Thats all.
The only hard fact what has something to do with our discussion here is the introduction of self-destruction at 8000 yards range into VT fuzes. This puts their effective range at 5000-7000 yards. And I have already explained why.
VT fuses in particular, although they failed a lot, change the equation.
A 5" shell has roughly 750 m/s velocity while the top speed of a torpedoe bomber at the time has roughly 125 m/s velocity.
At 8 km the flight time of a 5" shell is about 10.7 seconds and the maximum distance traveled by the plane is about 1.35 km. That does not sound great but the plane would not normally be flying perpendicular to the incoming fire. Keep in mind that the plane is there to do damage to a ship so it will be flown in a general direction towards a ship.
As an airplane's motion is primarily in the forward direction and would be flown generally towards a target most of the flight maneuvers would then be roughly linearly predictable. Therefore, one would expect the 5" shells to be in the general area of the attacking plane. This is where the VT fuse really matters because, instead of having to directly hit, the VT fuse makes it such that, on a coin flip (half of them failed to go off when they should), being within approximately 65 m of the attacking plane is good enough to damage it.
The time of flight going down with the distance of the attacking plane, and the opportunity for the plane to maneuver also going down if it wants to hit the ship, means that the whole system of systems gets better the closer the plane gets to the target. So your lower effective range of 5000 yards does not seem justifiable.
All that said, I am still gathering data on how HAA works in game and I am becoming convinced that it is quite complicated.
One thing I am noticing is that CAP is quite leaky, that is they let through a lot of stuff, at least for my planes and pilots.
5" shell time of flight.
4000 yards - 5.71 s
6000 yards - 9.67 s
8000 yards - 17.73
10000 yards - 20.81 s.
5" VT fuze radius of operation: 60, 70, 100 feets.
"The exercise found that the range patterns from USS Richard P. Leary were 260 yards at 6,000 yards (240 m at 5,500 m) and 470 yards at 12,000 yards (430 m at 11,000 m), which were both within the nominal accuracy specifications."
Torpedo bomber do not fly in the straight line right into immobile ship. It flies in the predicted torpedo launching point from there torpedo is expected to hit its moving target.
The actual tactic was:
"4. TORPEDO BOMBING. a. Daylight. Daylight torpedo bombing approaches are usually made in close formation at medium altitude. Attacks may be made in a wedge or loose diamond formation, or in small groups which separate to attack individual objectives from different directions. Glides are made at an angle of 40 to 45°, and torpedoes are dropped from an altitude of 200 to 300 feet at a range from 500 to 1,200 yards from the target. Approaches are planned from the direction where the least concentration of antiaircraft fire may be expected. Full advantage is taken of the position of the sun and cloud formations."
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So TBs were gliding from med to lo into predicted launch area ahead of the ship and from her perspective aircraft trajectory is a complex 3 dimensional curve while its motion changing from acceleration to slowing across the curve with the speed never being truly constant.
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Kamikazes.
"Typically forming up just outside 5-inch gun range for 10–15 minutes to select individual targets—sometimes allowing pursuing CAP to catch them—surviving kamikaze groups split up for final approach. High altitude attacks used the sun or clouds as cover, beginning a shallow angle (20–30 degrees) descent about 6,000–8,000 yards out. Some engaged in aerobatics. After closing to 1,500–2,000 yards, they engaged in terminal power dives up to 50 degrees, with little or no evasive maneuver, counting on speed to evade antiaircraft fire before impact. Low altitude approaches skimmed in 50–100 yards above the water; some closed to 1,000 yards before climbing sharply to 500 yards in altitude before diving sharply into their targets, other simply flew directly into the sides of ships.
Even with early warning and heavy CAP, kamikazes that penetrated to their target areas were often able to achieve tactical surprise due to a lack of coordination between shipboard fighter direction and search range and fire control radars. Incoming attackers were sighted equally often by lookouts and radar, but frequently too close to give effective warning. The average range Navy ships engaged kamikazes was 5,700 yards, while targeted ships did not open fire until 4,000 yards, well within the range of the 5-inch batteries and nearly at the effective range of the 20mm and 40mm guns.
The shortened range and reaction time diminished the effectiveness of the 5-inch guns and their lethal VT ammunition. Their fire control computers could not keep up with rapidly closing targets and the limited time for engagement reduced the number of shells fired. This left the burden of point defense to the 40mm and 20mm batteries. These guns were credited with 46.8 percent and 31.8 percent of the kamikazes downed by antiaircraft fire, respectively, while the 5-inch guns claimed 15.8 percent (VT fuzes accounted for 44 percent of that total)."
___
Curves are even more complex.
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"2. DIVE BOMBING. a. General. Japanese dive-bombing attacks, most frequent and effective in the early months of the war, are largely directed against shipping and equipment on beachheads. The accuracy of Japanese dive bombing is not outstanding and has been affected by Allied antiaircraft fire and fighter interception. Numerous reports make it clear that damaged planes, particularly dive bombers, attempt to crash on their targets as a last resort.
b. Formations. (1) The usual Japanese dive-bombing formations are in multiples of 3 as follows: 3-plane Vs in line astern; in 6-or 9-plane Vs; in Vs of Vs. The number of dive bombers employed varies with the nature of the target; for example, larger formations are employed against naval vessels than against merchant ships. Efforts are made to saturate enemy defenses by increasing the density of attacking planes. Of late, because of Allied fighter opposition, the approach to the target has been generally at altitudes of from 12,000 to 18,000 feet. Immediately before the initial dive, which is approximately one of from 35° to almost vertical (more often dives approximating 45°), the Japanese change their formations to one of loose echelon or string. Upon this change-over, the individual dives commence in rapid succession, usually from up-sun, from areas of restricted visibility, or from coordinates exposing them to minimum antiaircraft fire. The bomb release point varies from 500 feet to as high as 3,000 feet. This release point, it is believed, is governed by the pull-out point of the lead plane or the intensity of antiaircraft fire. It also has been noted that the bomb-release point is generally higher during dives approaching the vertical where greater speeds have been attained.
(2) When larger formations have been employed, Japanese dive bombers frequently divide their strength into smaller forces and attack a given target simultaneously from different directions."
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Time of flight to 12000 feets vertically is 6 s. To 18000 feets - 10 s. At 55 degree from horizontal - 8 and 14.5 seconds accordingly. Time of flight to the target at the horizontal range of 10000 yards and 18000 feet altitude is 30 s. So while dive bombers do their change-up ship is sending shells half-minute into the future what varies as in "maybe they will move further and go for the vertical dive or maybe they will already start more shallow one". May as well flip a coin.
This data is based on one battle with nine air attacks from the enemy fleet carriers(CV). The targeted fleet consisted of two Battle Cruisers (BC), two CVs, and two escort carriers (CVL) each with a mix of screens of destroyers (DD) and light cruisers CLs/CLAAs (CLAA has dual purpose primary guns). Eight of the nine air attacks targeted screens of the larger ships, one targeted a CVL. All ships in the sample have a single DP gun battery. The DP batteries of the ships in this sample are listed:
I will spare most of the details for further discussion but the most important discovery to come about by looking into how HAA works in detail in game is the overwhelming importance of the distance from the targeted ship.
The range of the ships from the targeted ship in the example air attacks I have described are summarized in this image:
https://steamcommunity.com/sharedfiles/filedetails/?id=3375980450
The average range of all ships from the air attack targeted ship is 3.755 km and the Std Deviation is 2.044 km but did not really follow a standard distribution except at the highest level of binning. The graph was generated by determining the ranges for every ship and binning them together in 0.5 km and 1.0 km bins. The data points are at the higher end of the binning range (i.e. the 1.0-1.5 km bin is shown at 1.5 km). The distribution of ships is roughly flat from about 2 km to about 6 km with only a slight dip off at 7 km.
Ship range information is not terribly useful by itself (more on this later) but can be corrected based on the DP guns maximum range along with noting how many ships contributed to the HAA score and at what fraction of their gun range.
https://steamcommunity.com/sharedfiles/filedetails/?id=3375985699
Note that I do not have enough data to be completely confident in the conclusions I am reaching from here on. However, it seems likely that the maximum fraction of DP gun range at which a ship can contribute to an air attacked ship's defense with HAA is 0.4. However, the maximum observed fraction of DP gun range contributing to air defense was about 0.365 (with good confidence for the accuracy of this fraction) so there is some uncertainty in the 0.4 fraction.
Note that the air attacked ship is not guaranteed to contribute to its own defense (about 90% of the time it does). The results of the above findings is that a ship with longer range DP guns at the same range has a higher probability contributing to the defense of a targeted ship with HAA.
This suggests: Probability_of_Ship_Assisting = 0.9-2.25*fraction_of_DP_gun_range
Further analyzing the data for what ships contributed to the HAA of ships being air attacked let me reach a preliminary partial table of the divisors for each ship by class and fraction of gun range.
https://steamcommunity.com/sharedfiles/filedetails/?id=3376021363
And in table form:
Divide a given ship's HAA factor by the divisor found by locating the class of ship and the fraction of its gun range to the air attacked ship.
These are important finding in three ways:
When combined all together it seems obvious that range is the chief consideration in determining the effective combined HAA factor in air defense.
Assuming a 3" DP armed CL were compared to a 4" DP armed CL, how much more would the 4" armed ship contribute to the HAA factor in the defense of the fleet.
Assuming the fleet distribution in my examples is similar to all fleets a 3" armed CL having 8 km max range would have approximately 69 out of 175 (assuming measuring from own ship this ignores the multiple ships at 0 distance) is 39.4% of the fleet within its HAA range.
Assuming a flat distribution (which is simplifying a bit).
Sum from n = 1 to n = 69: Probability_of_Ship_Assisting = 0.9-2.25*0.4*(n/69) = 30.6
This is the sum of probability of the ship helping the fleet assuming any ship in the fleet is equally probable of being any distance out to the maximum of 0.4*DP gun range.
Using the CL row with four quadrants of range for divisors.
(30.6/2+30.6/2.5+30.6/3+30.6/3.5)/4 * HAA = 46.48/4 * HAA = 11.6 * HAA
For a 4" armed CL with 10 km range using the same assumptions it is 102/175 (58.3% of the fleet covered):
Sum from n = 1 to n = 102: Probability_of_Ship_Assisting = 0.9-2.25*0.4*(n/102) = 45.45
(45.45/2+45.45/2.5+45.45/3+45.45/3.5)/4 * HAA = 17.26 * HAA
HAA factor on a CL with 4" DP guns is 17.26/11.6 = 1.488 ~50% more valuable to the fleet than the SAME HAA factor on a CL with 3" DP guns.
EDIT 4:30 12/1/2024 for fixing formatting and adding example calculation of relative value of different gun calibers.