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If you can send me a photo of the printed circuit card mounted inside the blower compartment of the furnace... showing where the thermostat wires attach to it, ideally so that I can read at least some of those terminal markings such as R, Y, W, G, C ...I can most likely debug the wiring error for you.
You can use the paper clip icon at the right end of the tool bar above for that.
The unit on the roof could be what we call a 'packaged unit' it contains both heating and cooling cooling equipment. If you can get me the brand and model number if it, I can look it up.
The web site charges an extra for a phone call... it is optional on your end.
pictures of roof mounted 'gas packaged AC units'
Picture of a cooling only or heat pump unit that requires an air handler or furnace inside the house, called a split system.
If you have a packaged unit on the roof, be careful not to fall off the roof, then remove one or two of the service panels on the side of the unit until you can see the large curvy blower inside... if the printed circuit card is not mounted there it will be behind one of the other panels... its about 6 x 8" square, more or less flat, and generally green colored.
That will work, this dialog box is all we need in most of these cases. There is no rush, I am in until 10 pm, then back all day tomorrow.
If you include a photo of the wiring diagram, printed on a sheet of paper and glued inside the same service panel that will help as well, it needs to be close up, well focused and taken straight on.
Rubber soled boots on the roof along with a rope tied to something on the roof that you can hang onto is a good idea.