I have a gas furnace and air conditioner. Trying to wire thermostat with only four wires available. Red, white, black, and green. Looked at my neighbours wiring and it seems I am missing an orange wire. Can I wire thermostat with only four wires? I have tried previously as follows: green-G, white-W, red-Rc, black-Rh. There is still a Y terminal open on my thermostat. The heating and fan work in heat mode but when on cool mode only the fan comes on but no ac. Circuit breakers to ac are in tact. Please help me with some higher knowledge.

Is your system a heat pump?

Unsure if heat pump? How can you tell.

You only have 4 wires which is usually a strait A/C type system and not a heat pump but I thought I would ask.

To start with you have NON standard wiring colors. Usual colors are white,red,green,yellow.

In your case try
Green to G

Jump RH and RC together and land the red wire together with the jumped terminals.

White to W

Black to Y

Since I mentioned before that you have NON standard thermostat colors for the wires the above is just a good guess on my part.

You may have to trace down each wire to see where it goes if the above does not work.