The answer to 'why is my P-trap constantly running' is that a P-trap that is constantly running or draining water is likely due to a clogged or damaged trap. The P-trap is the curved pipe under your sink that is designed to hold water and prevent sewer gases from entering your home. If the trap is not working properly, water can continuously drain out, causing the trap to run constantly.
Some common causes of a constantly running P-trap include a clogged drain, a broken or cracked trap, or a vent pipe issue. If the drain is clogged, the water has nowhere to go and will continuously drain out of the trap. A cracked or damaged trap can also cause water to leak out. Additionally, issues with the vent pipe, which allows air to flow through the plumbing system, can disrupt the water seal in the trap and cause it to run constantly.