Player Capital -> Gift City [20 TILES] -> AI Capital [20 TILES]; Result = Can't (Liberate).
So in case of a tie, (Liberate) is not an option any longer.
Player Capital -> Gift City [21 TILES] -> AI Capital [20 TILES]; Result =
Can (Liberate) city to AI.
So OurDistance > HisDistance * Modifier?
If True, then (Liberate) is possible.
Just have to figure out the correct culture modifier.
Gift City has 1

for the human player:
Player Capital -> Gift City [21 TILES] -> AI Capital [20 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [19 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [18 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [17 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [16 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [15 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [14 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [13 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [12 TILES]; Result =
Can (Liberate).
Gift City has 2

for the human player:
Player Capital -> Gift City [21 TILES] -> AI Capital [12 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [11 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [10 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [9 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [8 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [7 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [6 TILES]; Result =
Can (Liberate).
Gift City has 3

for the human player:
Player Capital -> Gift City [21 TILES] -> AI Capital [6 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [5 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [4 TILES]; Result =
Can (Liberate).
Gift City has 4

for the human player:
Player Capital -> Gift City [21 TILES] -> AI Capital [4 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [21 TILES] -> AI Capital [3 TILES]; Result =
Can (Liberate).
Inch the player capital 1 tile closer and run it again.
Gift City has 1

for the human player:
Player Capital -> Gift City [20 TILES] -> AI Capital [20 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [20 TILES] -> AI Capital [13 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [20 TILES] -> AI Capital [12 TILES]; Result =
Can (Liberate).
Gift City has 2

for the human player:
Player Capital -> Gift City [20 TILES] -> AI Capital [7 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [20 TILES] -> AI Capital [6 TILES]; Result =
Can (Liberate).
Gift City has 3

for the human player:
Player Capital -> Gift City [20 TILES] -> AI Capital [5 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [20 TILES] -> AI Capital [4 TILES]; Result =
Can (Liberate).
Gift City has 4

for the human player:
Player Capital -> Gift City [20 TILES] -> AI Capital [4 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [20 TILES] -> AI Capital [3 TILES]; Result =
Can (Liberate).
So weird, the results are exactly the same even though the player capital was 1 tile closer.
1 more tile closer!
Gift City has 1

for the human player:
Player Capital -> Gift City [19 TILES] -> AI Capital [20 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [19 TILES] -> AI Capital [13 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [19 TILES] -> AI Capital [12 TILES]; Result =
Can (Liberate).
Gift City has 2

for the human player:
Player Capital -> Gift City [19 TILES] -> AI Capital [7 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [19 TILES] -> AI Capital [6 TILES]; Result =
Can (Liberate).
Gift City has 3

for the human player:
Player Capital -> Gift City [19 TILES] -> AI Capital [5 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [19 TILES] -> AI Capital [4 TILES]; Result =
Can (Liberate).
Gift City has 4

for the human player:
Player Capital -> Gift City [19 TILES] -> AI Capital [4 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [19 TILES] -> AI Capital [3 TILES]; Result =
Can (Liberate).
Still the same?
That can't be right.
So one more test.
Gift City has 1

for the human player:
Player Capital -> Gift City [18 TILES] -> AI Capital [20 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [18 TILES] -> AI Capital [13 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [18 TILES] -> AI Capital [12 TILES]; Result = Can't (Liberate).
Player Capital -> Gift City [18 TILES] -> AI Capital [11 TILES]; Result =
Can (Liberate).
Ugh, can't see it. (The right math formula)
Did I mess up a test somewhere and get wrong data?
It seems like 0

has a 1 factor, 1

has a 1.5 factor, 2

has a 3 factor, 3

has a 4 factor, and 4

has a 5 factor.
But that is not quite right.
A gift city with 1 culture that is 21 tiles from the player capital will not (Liberate) unless the AI capital is 12 tiles or closer.
With a 1.5 factor, I would expect 13 tiles or closer.

Tested it twice.
I'll attach a test game so someone can play around with worldbuilder.