Adam,
The outside temperature display cable at the instrument cluster does have 4 wires (green and brown for the sensor and 2 others for power). However, I'm quite sure that your cut orphan cable is something else.

Almost certainly your problem is a dead sensor - which can be easily replaced for less than $5. When the thermistor fails, causing the resistance value to fall outside the display range, you will get a blank display.

Earlier I posted the diagnostic and replacement procedures for the sensor, as well as the relevant part nos. I suggest you test your sensor cable with a resistor as described (and leave that orphan cable alone).

Let me know if you have difficulty finding the procedures or have any further questions.
Good Luck, Barrie