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Shouldn't this perhaps get mentioned in the project, like "Our engineers/researchers have given up on solving the complexity behind this project" or something?
I believe that having that option is one of the settings you can make in a custom campaign.
It has two chances it rolls. One is the overall unlock chance, which is what's shown on the tech tree, and is affected by how much of the unlocking global research you contributed to. If that rolls a failure then it won't unlock. If it rolls a success, then a monthly chance will start firing to make it actually available to you, which has a base chance, a monthly growth to that chance, and a cap on the monthly growth.
The first 3 and the base are included into the % shown for the project under its requirements. The contribution bonus is not shown but is added at the moment the tech unlock roll occurs. Many techs start at 100%, and most you can easily reach 100%.
Provided you pass that first check, the projects then get a starting monthly unlock chance (which actually occurs as a daily check spread across the month). This is increased each month by a monthly unlock chance increase (again, normalized over the month) until it hits the maximum monthly unlock chance. These values are specific for each project, and for some can be quite low which means it may take a very long time to actually unlock.
This starting monthly unlock value is increased by your contribution bonus and your council Sci bonus, and can actually exceed the maximum monthly unlock chance but will then never grow. The ARG project increases the monthly increase by 1%.
So, if you have a project with a base unlock chance of 10%, have one councilor with Innovative, the sum total of your councilors' Sci stat is 72, you have the ARG project done, and you contributed 23% to the globals, your initial unlock roll would be 10+10+12.4+10+23 = 65%. Provided you passed that check, and the project's starting unlock base was 20%, increase was 5%, and maximum was 80% your monthly check would start at 20+23+12.4 = 55%, and it would grow by 6% each month until it hit the cap of 80% or you unlocked it, whichever happens first.