For each phase lane jump away your planet is from your capital it loses about 10% off its maximum allegiance cap. Culture can't fix this, unfortunately. The only solution is to move your capital closer (Advent also have a late-game research that can boost allegiance by 10%)

All other things being equal however, allegiance will never get lower than 25%. So you're pretty close to the bottom.

There's no way to increase the rate of research (aside from altering it in the start-up options) except for an Advent tier 1 research.

It won't surpass the cap. Your solution is only valid if his low allegiance is due to opposing culture.

"Capitol" is the building where the legislatures of western democracies meet. '

I.E. Westminster is the capitol of the U.K.
Capitol Hill is where the U.S. Congress meets.
The Reichstag Building is the capitol of the German Federal Republic.

I have never, ever seen this.

The increase in allegiance via culture is 10% above the "default" amount for a given distance from your capitol.

Multiple culture towers are/i] useful: it allows you to increase the spread of your culture even further -- usually when acting against hostile culture -- and to a limited degree, faster.

One culture center can stop two or three, but add in the culture upgrades and things get [i]really dicey, even to the degree of totally halting any enemy advance by denying them the ability to colonize the planet, even if they take out its own turrets. (I'm well known for my culture-spamming habits, for some reason...)

Iirc TEC gets Anti-matter regeneration in friendly culture and Advent get shield mitigation, So if you culture bomb a planet and then invade your taking less dmg to your shields + the ability of your support ships that knocks dmg off the top. coupled with Advents high shield tech. ( i forget what Vasari bonus is in culture... dmg maybe?)

Also you can use the Culture cannon thing to wipe out enemy culture in a system, jump in Nuke the planet, colonize it and then set up Culture spreading buildings to secure your hold in the region

The home world starts with 100%.
Every jump costs 10% base allegance, so a planet 4 jumps away would have 60% base.
If the planet is under friendly culture (your own or your allies), this gets raised by 10%.
It does not matter how this culture is produced - Advent super weapon, 1 broadcast center, 10000 broadcast centers. It's always 10%, not more, not less.
Minimum possible is 25%. So if you play on a giant map were all your planets are far away from home then they will all have 25%. Build a broadcast center every 3-4 planets and they will rise to 35%. If you just want the 10% increase then building more broadcast centers is a total waste, besser go for trade ports.
Defense against enemy culture is easy. You need less defensive broadcast centers to hold a planet so to invade your empire with his culture, the enemy has to waste a LOT of logistic slots and you can counter by just having less then half the broadcast centers he has, leaving you with a much stronger economy (since the other slots can be used for trade ports).
Advent lategame tech will also increase the base value by 10% on all planets, making it one of the best income enhancing techs on large maps.

On large maps it also helps to place your home planet directly next to a star. This way, all planets in other systems will be closer, starting with a higher base allegiance value.

I hope this sums it up