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2) research available techs that add extra research options
3) take perks that add research options
4) pray harder to RNGesus
(I'm assuming you have scrolled down to make sure it isn't in the partially researched list)
Cruisers has a pretty significant penalty to availability before 2220 as well so that could potentially be your issue
He said he had all the destroyer techs... advanced destroyer hulls is the same tech tier as cruisers...
As the game's year count increases, certain Techs get RNG draw weight *multipliers*, not RNG draw weight *divisors*.
Cruiser tech has a .1 weight modifier before 2220 in addition to the x2, x3, and so on it gets at later dates. Basically it's set up so that nearly everyone gets it (or at least has the option) in a fairly narrow window. Battleships have a similar mechanic except the year is 2250...
https://stellaris.paradoxwikis.com/Engineering_research (click expand and read the modifiers...)
and lowers all others in that category, so by the time everyone reaches their final break through, everyone is unique in their sciences, and is urged to use diplomacy, to learn others from other nations.
that mod handles RNG well, because its at least SOMEWHAT in your control.
I didn't say the window gets more narrow over time, you might want to brush up on your comprehension skills...
I said the system is set up so that it's likely everyone first gets access to it within a fairly narrow window. This makes sense from a balance perspective-don't let anyone have exclusive use of larger hulls for very long as it can be a significant advantage.
So until 2220 you are unlikely to get the card due to the .1 weight modifier. At 2230 the weight gets doubled, 5 years later it goes up to 6x, 5 years later it hops to 24x, plus it goes up by 10x if a neighbor has it so by 2240 the weight is 240x normal if you border someone with the tech, or 2400x more likely than in 2219 without a neighbor. Thus the vast majority of nations will see the card for the first time within a 20 year window and then likely see it every other draw (or however often it is eligible) until they learn it.
You then see the exact same scenario with battleships starting in 2250. Titans have a slightly wider but similar set-up as well.