Then I recommenced chewing. Per aspera ad astra. The mundane truth is that gum disintegration is caused by carbohydrate digestion. Saliva contains an enzyme called ptyalin that breaks down the branched polysaccharides of the carbohydrates into oligosaccharides. In other words, if you hold thoroughly masticated gum in your mouth long enough without chewing it, the saliva begins digesting the gum's carbohydrate linkages.
Physiology solves another enigma.
The disintegration of chewing gum while exercising, eating butter or other food, kissing, or engaging in more expressive demonstrations of love or lust is not in the least mysterious. Nor is there any need to invent exotic digestive hormones. Given time, the enzymes in ordinary saliva at body temperature will cause chewing gum to break down. However, the oral cavity is rarely at body temperature, since it's cooled by the passage of air during breathing. In addition, chewing tends to knock the enzymes loose from the gum before they can do their stuff.