Why does the anti-theft light stay on?

There is a problem with the system an there is a DTC - diagnostic trouble code stored in the body control module !
The vehicle theft deterrent system is designed in order to prevent vehicle theft by disabling the fuel injectors unless the lock cylinder is engaged by the correct mechanical key. The theft deterrent system uses the following 4 components for theft prevention:
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The lock cylinder

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The ignition switch

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The body control module (BCM)

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The powertrain control module (PCM)

When starting the engine, the PCM searches for a password from the BCM through the Class 2 serial data circuit. If the password is not recognized or not present, the PCM will disable the fuel injectors. Two modes of tamper detection are provided:
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No password received

The engine will start and stall quickly. SECURITY telltale will flash on the instrument panel cluster (IPC) and then stay ON steady.

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Incorrect or disable password received (More than 3 invalid passwords are received)

The engine is disabled for at least 10 minutes and the SECURITY telltale will illuminate solid on the IPC during the 10 minutes.

After the vehicle has passed theft detection, the PCM will continue normal operation
Ignition Switch
The mechanical key and lock cylinder is located on the steering column. The main electrical switching portion of the assembly is separate from the key and lock cylinder. The VTD portion of the ignition switch is the Passlock?"¢ cylinder.
The Passlock?"¢ lock cylinder is a locking cylinder that turns a magnet past a stationary hall effect sensor. When turned to crank, the hall effect sensor is latched which applies a fixed resistance to the Passlock?"¢ signal data circuit. The voltage measured by the BCM on the Passlock?"¢ signal data circuit is the Passlock?"¢ data value.
Body Control Module (BCM)
The BCM compares the Passlock?"¢ data value to a value that it has stored in memory. If the values match the BCM sends a password via the class 2 serial data line to the PCM.
Powertrain Control Module (PCM)
When the PCM receives the correct password, the PCM enables the fuel injection system, allowing the vehicle to operate correctly. The PCM may allow the car to start and quickly stall during a failed theft condition.
If the BCM does not receive a Passlock?"¢ data value, the BCM will enter a short tamper mode. During this mode, the VTD system will not allow the car to operate for 4 seconds.
If the Passlock?"¢ data value is incorrect, the VTD system will enter the long tamper mode. In the long tamper mode, the following actions will occur:
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The THEFT SYSTEM indicator will flash.

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The fuel injectors will be shut off for approximately 10 minutes.

Although the vehicle may start, the engine will quickly stall due to a fuel cut-off.
In the event of an open in the serial data communication between the BCM and the PCM or a failure in the Passlock?"¢ lock cylinder, the following actions occur:
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The VTD system will become fail-enabled if the car has already passed theft for that ignition cycle (i.e. the engine is running).

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The PCM or the BCM may set diagnostic trouble codes (DTCs).

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The THEFT SYSTEM indicator will light continuously.

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The PCM will become fail-enabled for future ignition cycles.

If a failure occurs when the ignition is off, while the VTD system is not fail-enabled, the PCM will never receive a valid password, and will not allow vehicle operation by disabling the fuel injectors. An this system can set 6 DTC'S
DTC B2947 Security System Sensor Power CKT Low DTC B2948 Security System Sensor Power CKT High DTC B2957 Security System Sensor Data Circuit Low DTC B2958 Security System Sensor Data Circuit High DTC B2960 Sec Sys Sensor Data Incorrect but Valid DTC B3031 Security System Controller in Learn Mode DTC B3033 Security System Indicates Tamper