answer:First off, I don’t know what “4 seconds every 24 hours” means, but I’m assuming that your tank is filling four times every day, even when the fixture isn’t used. If we accept the results of the dye test and believe that no water is leaking past the flapper into the bowl (still the most likely path) then we still have to find “where does the water go?” If this is really such a mystery, then I’d want to verify that the tank actually fills properly when the flushing mechanism is activated. That is, it could be possible that the float arm is set to such a point that the tank hardly fills at all. (Unlikely, I suppose, but I’d want to see the mechanism actually work a couple of times to know that it does work properly.) If the tank is actually filling, and if the flapper valve is the only path for water to take to the bowl (because it doesn’t have to be, but should be the only path), and if you don’t have gremlins or pranksters for roommates (the pranksters being the more likely) then there’s a leak in the system somewhere. Since there’s no water apparent on the floor, then the tank itself appears to be intact. My guess is that the overflow pipe itself has a leak. Toilet tanks have an overflow standpipe so that a faulty float or shutoff valve won’t flood the place. If the tank is overfilling, then the water runs into the top of the overflow and out the drain. Depending on the design of the toilet, the overflow should bypass the toilet bowl and go straight to the drain (so that the combination of a blocked toilet and bad float / shutoff won’t flood you). If you can, remove that overflow pipe and examine it for pinholes or cracks. And welcome to ask-public, btw. Stay dry; stick around.