Use key history, battery-voltage context, module communication, and key-programming clues to separate a wrong-key event from a programming problem or a failing immobilizer path.
A new key, dead battery, module replacement, jump-start event, or water intrusion matters here more than usual. Immobilizer complaints often follow configuration or power-history changes, not just random part failure.
P0513 usually leans toward authentication failing in real time, while P0633 more specifically points toward a key or PCM relationship that was never programmed correctly or was lost after module work. Both can create a no-start, but the repair path is not identical.
A clean crank-no-start with a security light often means the PCM is not the only module you need to scan. Body, theft-deterrent, steering-column, or gateway modules can hold the real reason the key is being rejected.
P0513 often fits a car that suddenly says the key is wrong today, especially if a spare key behaves differently or the theft light changes immediately during crank. P0633 fits better after PCM replacement, key programming, or lost memory, where the hardware may be fine but the learned relationship is missing. Both can end in a no-start, but one is an authentication event and the other is often a configuration story.
Low battery voltage and unstable grounds can corrupt authentication, wake-up timing, and module communication badly enough to look like a key problem. Prove basic power and ground integrity before replacing expensive anti-theft parts.
If the complaint started after module or key work, programming rises to the top. If it is intermittent and follows voltage or communication issues, widen the diagnosis beyond the key itself. The winner is the branch you can prove with module data, not the one with the most dramatic dashboard message.
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Built to keep related fixes, symptoms, and hubs one step away.