After much dismay and fumbling about (10+ hours of research and logic), here are my discoveries, and the logical arguments they have brought me down:
1) All large weapons are required to win the Battleship duels (at the cost of worsening performance against Cruiser stacks)
2) Because of how focus-firing works, Tachyon lance is required.
3) From 1 and 2: Ergo, we now know the mandatory ship pieces: Spinal Mount / Artillery Core / Artillery Stern
1) Because of how focus-firing works, an all Kinetics build is not viable: the damage gets too spread around.
2) From 1, 0 Shields / 5 Armor is not an optimal Battleship, because it would be optimized to beat an All Kinetics build, which we already know is bad.
3) From 2, you have to run some shields, so you need 2 Shield Capacitors.
4) 4 Shields / 1 Armor is not an option for shield/armor configs because of power limitations (although 4 Shields / 0 Armor is possible)
5) From 2 and 4, the only shield/armor configs that are viable are: 4 Armor / 1 Shield, 3 Armor / 2 Shield, 2 Armor / 3 Shield, 0 Armor / 4 Shield
Now, for each armor config, notice that here is a corresponding weapon set that optimally defeats it:
1 Shield / 4 Armor: 4 Plasma
2 Shield / 3 Armor: 3 Plasma, 1 KA
3 Shield / 2 Armor: 2 Plasma, 2 KA
4 Shield / 0 Armor: ??? 1 Plasma, 3 KA ???
As the number of KA increases, the focus-fire potential decreases. Hence, let's consider ruling out the 1 Plasma, 3 KA attack rig.
In 10 match-ups of a 1 Shield / 4 Armor build (which defends optimally against 2 Plasma / 2 Kinetic) with 1 Plasma / 3 Kinetic, this build still lost to a 4 Shield / 0 Armor build with 2 Plasma / 2 Kinetic. This leads me to believe that 1 Plasma / 3 Kinetic is not a very viable weapon set, as it should have won here because it's supposed to hard-counter 4 Shield / 0 Armor.
Now, it's possible here that because there's no good counter to it, (we're now ruling out 1 Plasma, 3 KA) that 4 Shield / 0 Armor is the best possible defense build.
Let's see how it performs against a 3 Shield / 2 Armor build when both run 2 Plasma / 2 KA (the best weapon set against 3 Shield, and the next best against 4 Shield).
Aaaand after 10 match-ups, the 4 Shield / 0 Armor build wins ~80% of the time, with far more survivors afterwards as well.
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So what does this mean? It means the 4 Shield / 0 Armor build is simply the best possible shield / armor config.
Why?
A) It almost beats the opposite shield / armor config, even when using the worst weapons against it (2 Plasma 2 KA against the 1 Shield / 4 Armor build)
B) It beats its direct competition config using the same weapons against it (2 Plasma 2 KA against the 3 Shield / 2 Armor build)
Given the 4S / 0A config is the best, what's the best weapons?
Well, that's going to depend on the defense config of your target Battleships. If they run 1 Shield / 4 Armor, it's definitely 4 Plasma.
However, if they're smart and also run the 4S / 0A config, there will be a simple fact about which weapon set is best. 4 Plasma is obviously bad against 4 Shield, but let's try the remaining three options.
Designating the name of the build by the number of shields, and then the number of plasma, we have limited the build options down to:
43
42
41
So there's three match-ups left to empirically check:
43 vs 42 : the 42 CRUSHES the 43. Not even close.
43 vs 41 : the 43 solidly defeats the 41. If 41 gets lucky it can clinch a victory.
42 vs 41 : the 42 solidly defeats the 41. If 41 gets lucky it can clinch a victory.
Thus, the 42 is the best Battleship in the game right now.
Here's what it looks like:
Now, the 12 can beat the 42. However, if the 42 re-outfits to a 44, the 12 will lose every time.
Someone may also run a 22. Re-outfitting to a 43 will win with strong frequency.
Basically:
44 > 12 > 42
43 > 22 > 42
42 > Everything That's not 12, 22
Conclusion: the 42 is overall the best Battleship, because unlike a 0X, 1X, 2X or 3X build, a 4X build has no weapon set that hard-counters it. 2 Plasma / 2 KA works best, but it's not a hard-counter. Among the 4X builds, the 42 is the clear winner.