Combine the oils and vodka. Slowly whisk in the gum arabic. The mixture will turn cloudy yellow in color. If the oils and vodka still separate when the mixture rests, add a small amount of gum arabic until the mixture is emulsified.
Make the Cola Formula
In a jar with a lid, infuse the bay leaves, basil leaves and vodka. Let sit for 48 hours. Reserving the liquid, strain out the leaves.
Score vanilla pods lengthwise with the tip of a knife. Using a spoon or knife, carefully scrape the seeds from the pods. Discard the pods or save for a later use.
In a pan over medium low heat, warm the vanilla seeds, sugar and water. Stir until sugar is dissolved, about 5 minutes. Stir in the honey. Heat for 25 minutes more. Strain the liquid through a fine mesh strainer or a cheesecloth.
In a large bowl, combine 3 1/2 ounces of the vanilla syrup, the lime juice, citric acid, Cola Flavoring, citric acid and caffeine powder, and 1/2 teaspoon of the basil and bay leaf infusion. Stir until combined.
Stir the mixture back into the vanilla syrup. Decant into a glass bottle and store in the refrigerator for up to 1 month.
Make the Cuba Libre
In a highball glass filled with ice, add rum, cola formula and soda water. Garnish with a lime wedge.