Toilet is one year old and started adding water every fifteen minutes or so. Replaced flapper and the problem continues.
Yeah, your new flapper doesn't seal. That happens, especially if the flapper is shaped or rigid plastic. I had the same thing happen to me a year or more ago with a rigid plastic flapper (with large rubber washer). I closely examined the seat for burrs or gaps, but it was smooth and unbroken, so the problem was clearly a flapper misfit. If you do find a burr, I recommend that you gently file the burr away using a fine wood file.
I've found the best replacement flapper to be flat and made of flexible rubber. This will bend just enough under the tank's water pressure to seal against a seat that's not perfectly flat. Shaped flappers (like a mushroom) are intended to sit inside the opening like a round peg in a round hole often turn out not to be positioned perfectly to drop exactly into the hole. But a flat flapper won't behave badly like this if its hinge is not positioned perfectly relative to the hole.
Of course, also make sure your float isn't being held underwater after the flap closes. That's likely to cause the flap to lift and leak.