I like your mathematical approach to this. But I'm afraid there are just too many variables in the next 1000+ years to be able to account for everything.
Sure, Elite is fundamentally a fictional setting, but it's interesting to consider how the technology could work if was a real-life firearm design.
For example, let's build a Class 3 multicannon using an existing weapon as a starting point, walking through the design process as if it were a real firearm.
Comparing the Class 3 Multicannon to the 40 mm Bofors
We're gong to start with this weapon:
And try to create this:
Design Comparison
The best comparison we have as a starting point using existing autocannon designs would probably be the Bofors L/70 40 mm autocannon in terms of overall size and projectile weight. The main advantage of this comparison is that there are two versions of the Bofors, the L/60 and the L/70 version, which gives us some idea of how to scale up the design in terms of expected performance. Additionally the Bofors has seen widespread use as an anti-aircraft gun, a naval mount and is even carried by the AC-130 spectre gunship in an air-to-ground role so it's a weapon that has been used effectively in a variety of different roles and mountings.
Dimensions and Weight
The first issue to address is the barrel length of the Class 3 multicannon. Due to the extending barrels of the Class 3 multicannon it's not possible to get an exact barrel length but we can get a reasonable visual estimate based on the size of the hardpoint. The Class 3 large hardpoint dimensions are approximately 7.5 m long when stowed and approximately 10 m long when desployed with barrels extended. The barrel length for the Class 3 multicannon when extended is approximately 1/2 of the total length of the weapon which works out to approximately 5 m. The L/70 Bofors version uses has a barrel length of 70 calibres, i.e., 40 mm x 70 = 2.8 m. A 5 m barrel on the Class 3 mutlicannon would therefore be the approximately equivalent of a L/120 barrel in comparison. This is a relatively long barrel for an autocannon but the comparison does allow us to scale the weapon up reasonably well from the existing Bofors L/70 design.
Regarding total gun weight, the L/60 Bofors weighs 1981 kg or approximately 2 tons while the L/70 Bofors weighs 5150 kg or approximately 5 tons. Giving our assumption of 25% ammunition storage for a 8 ton weapon mount, that leaves 6 tons for the Class 3 multicannon. The additional barrels would increase the weight of the design compared to the single-barrel Bofors designs but a 6 ton weapon with 5 barrels of L/120 length would be reasonable with modest weight-reducing strategies. We can currently achieve significant weight savings in existing firearm designs by replacing steel components with lighter materials such as titanium, for example, the M134D-T minigun design which uses this approach. It would be reasonable to keep the Class 3 multicannon design within a 6 ton weight limit assuming similar weight-reducing strategies were employed.
Projectile Weight and Muzzle Velocity
The projectile weight for the L/60 Bofors is 900 g for the high explosive 40x311 round at a muzzle velocity of 880 m/s. The L/70 fires a longer 40x364 round which could propel a similar projectile at a higher muzzle velocity of 1030 m/s. The muzzle velocity of the L/70 is similar to that of the M61 Vulcan's 20 mm rounds or the GAU-8 Avenger's 30 mm rounds so again this scales well with previous comparisons, again assuming that we're keeping the muzzle velocity constant and simply increasing the projectile weight.
If we wanted to estimate the advantages of a longer barrel length in the Class 3 multicannon, we can compare the muzzle velocity for the L/60 and L/70 versions of the Bofors design. In this case the muzzle velocity increases from 881 m/s to 1021 m/s from the L/60 to the L/70, giving a muzzle velocity that is roughly proportional to barrel length. With an L/120 barrel on the Class 3 multicannon we could reasonably expect that the considerably longer barrel would generate a muzzle velocity of around 1600 m/s, i..e, approximately double the length and muzzle velocity of the L/60 barrel, which would match the in-game projectile velocity.
Rate of Fire
In terms of rate of fire, the L/60 Bofors has a rate of fire of 120 rpm while the L/70 version has a substantially higher rate of fire of 330 rpm. The class 3 multicannon has a rate of fire of 6.7 rounds per second or approximately 400 rpm. It wouldn't be difficult to achieve a fire rate of 400 rpm with a 5-barrel gatling design so that is not a difficult issue to scale up as we're only increasing the ROF by approximately 20% over the Bofors L/70. The 5-barrel design, especially if it is externally powered, could also provide considerable improvements in sustained fire and reliability over the single Bofors L/70 gun.
Ammunition Weight
The cartridges for the Bofors weigh approximately 4.8 pounds, or 2.18 kg, which means the 900 g projectile weight is approximately 41% of the total cartridge weight, close to the 50% assumption I had been using in my prior post. This means an ammunition load of 2100 rounds would weigh over 4.5 tons, which is over 50% of the weapon weight, and that would leave less than 4 tons for the gun itself. This is an issue in terms of fitting both the gun and the ammunition load within the 8 ton limit, but it's not insurmountable even with modern technology if we use caseless rounds, which can reduce of the total cartridge weight substantially compared to conventional rounds. If we assume a very efficient caseless design combined with modest advances in propellant it could be possible to design a 1 kg round where only 10% of the total cartridge weight consists of a highly efficient propellant that propels a 900 g round. This would be necessary to get the weight down to 1 kg per round or approximately 2 tons for 2100 rounds of ammunition. It would also keep the weapon's ammunition storage and feed systems more compact as well, which would be another advantage.
Performance Comparison
Essentially in comparison to the 40 mm Bofors L/70, the gatling design of the Class 3 multicannon fires rounds with approximately 2.5X more muzzle energy than the single L/70 gun, and also achieves 1.2X the fire rate due to the gatling design. That means that the combination of the increased fire rate (due to the gatling design) and the higher muzzle velocity (due to the longer barrel length of 120 calibres) would create a weapon that is 3X more powerful and effective than the single L/70 Bofors, while being very similar in overall weight and dimensions.
We do know, however, that the stock Class 3 multicannon design in Elite is not at its design potential. If we push it further to the current Engineering limits with a Grade 5 overcharged modification, i.e., using rounds that are approximately 45% more powerful while maintaining a similar ROF, we can produce a weapon that is nearly 50% more effective. That now puts an overcharged Class 3 muticannon at approximately 5X the overall effectiveness of a single L/70 Bofors given the combined increases in damage, rate of fire and muzzle velocity.
Additionally, in terms of overall weapon effectiveness there's the argument that the explosive contained in the Elite autocannon rounds could be substantially more powerful and could use armor-piercing materials which are more advanced than current rounds. There's also the issue of using experimental rounds such as incendiary or corrosive rounds as well. Since we have no way of measuring or quantifying this, however, it's purely speculative and we don't really have any in-game comparisons to use to assess this properly. This could, however, explain in-game effectiveness that was substantially higher in comparison to a modern round of similar calibre and muzzle velocity. It is worth noting that when combined with the design advantages of a Class 3 multicannon that already provides approximately 5X the effectiveness over a single Bofors L/70 gun, these factors could produce firepower that is even more effective than the above comparison would suggest, and this could result in a gun design that is 10X (or more) effective than the single Bofors L/70.
Caveats and Limitations
An argument could be made that a Class 3 multicannon could use stronger and lighter materials or manufacturing technologies to achieve substantially higher performance in certain areas. Conversely, there's the issue that the weapon needs to operate in a vacuum, at extreme temperatures and under high g-forces during ship maneuvers. Additionally we're assuming that the weight of the gun includes the servos and tracking system that is built into the weapon mount for a gimballed weapon. If we take the gun comparison at face value it would seem reasonable given current technology to make the basic design work within the weight and perfomance limits. Assuming that improvements in materials and design would be balanced out by the dramatically harsher and more demanding operating environment that the gun would be subject to in space would make this an issue that we wouldn't necessarily need to address as part of our comparison. In other words, whatever advantages the design gets in terms of more advanced technology would be used to allow it to operate at a much higher level of reliability in a much more demanding operating environment and that is really a different discussion. It could also explain why Elite guns would be bulkier than modern weapons of similar weight, i.e., they may require more rugged or bulkier designs with thicker barrels or other special design requirements compared to modern firearms while achieving much higher overall performance at a similar weight. In particular, the apparent size of the thicker barrels in Elite compared to the calibre of the ammunition would actually make sense given that the barrel would need to withstand much higher operating pressures to achieve the higher muzzle velocities involved and would likely be heavily overbuilt and engineered to operate effectively and reliably in space.
The overall conclusion here is that using known parameters for weapon weight, calibre, barrel length, muzzle velocity, ROF and ammunition storage could result in a practical and effective firearm design. There would be additional factors to consider in terms of the weapon's performance given the operating environment and other design issues but the core design features would be reasonable and effective based on current principles of firearm design.