My toilet tank fills about 4 seconds every 24 hours.

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 Fluther, btw. Stay dry; stick around.

Your toilet is dripping, let’s say just 10–15 drops/hour. In 24 hour it loses 240–360 drops. That will make a good 2–3 deciliter water. 4 seconds are just enough to replace that water, and 10–15 drops is the amount what you can’t notice in one hour unless you sit by the toilett for one hour with a towell in your hand pressed against the inside of the porcelan…

Put a couple of drops of blue food coloring in the tank and a drop of red coloring in the over flow pipe. It will be readily apparent which one is the culprit. A leaking flapper will turn the tank water blue in a few hours. A leaking overflow will either not show up at all or turn the tank red depending upon which kind of toilet you have.
This will give you a place to start.
If your floor turns purple you have bigger problems.

Your flapper is not doing its job, inside your water tank. it is leaking water into the bowl, which causes the bowl ball to drop, which in turn lets more water into your water tank to compensate for the water leakage.

You need to replace the internal parts of your water tank at your commode.

Home Depot has everything you need. i just did this myself.

Oh, I get it now: a 4-second fill every 24 hours. Yeah, now that makes sense.