I will guess that one of the wires to the alarm switch has failed in the door harness. the alarm switch has two 4.7K ohm resisters in side it. when the alarm is deactivated the current only runs through one of the resisters. when the alarm is armed the current runs through both resisters in parallel (2.4K ohms). the system is designed so that if the module doesn't see the single resistor 360 ohm value it triggers the alarm. since your system was not armed using the key I can only assume that the door harness has an issue. either a broken wire or bad insulation that breaks connection or shorts the alarm switch wires. the module sees the out of range low (shorted wires) or high (broken wire) resistance and triggers the alarm.
for reference here is a thread I posted recently on
how I made an alarm key. it has a couple pics of how the alarm switch is constructed.