Car will not start (does not recognize key) - 2006 Infiniti FX35

There's an antenna for the transponder receiver built into the key surround on the 9-5, it could have had a knock or be faulty.

On the 9-5 if the transponder stops working and you get a key not accepted message then you can override it by pushing the unlock button on the key remote with the ignition turned to on and it should let you start the car.

Hold your old key together with your new key and start the car. After a few seconds turn it off, remove the old key and restart it. If it starts, the car would recognize your new key.

The key codes work in sequence. Each time you press a button on the key, it generates a new code. The car knows what code is next, and the few that come after.
If you accidentally press the button when the key is out of range of the car, the code the key sends isn't recognized by the car as "current". The next time you press it, the car does recognize it, as it's re-synchronized.
It's possible that the battery in the key is getting weak, and that is simply time-related. Car batteries have a similar lifetime. I'm a little surprised that your alternator went out, too, but that does happen.

Maybe this will help:
From the owners manual:

SENTRY KEY IMMOBILIZER - IF EQUIPPED
With this system, an electronically coded ignition key
sends a signal to the vehicle electronics. If the electronics
recognizes the signal, the vehicle will start and continue
to run. If the system does not recognize the signal, the
vehicle will start and run for 2 seconds, then shut off.
After six unsuccessful attempts at starting, the system
will shut down until the correct key is used.

That's exactly what happened when I cut a new key and tried to start the truck without programming the key. So it looks like your truck is not recognizing the key.
If you have another key, try that one.

Or, try and disconnect the battery for a while, that clears up a lot of electrical problems.

GL,
Td

Wait for the new key, honda is the only one that can deal with it unless you can find a lock smith that deals with those.

there is a keyless entry control module

Yes the your Lexus does have a physical key slot to start the car

MAKE SURE YOU DONT HAVE THE WRONG KEY.TRY ANOTHER KEY.PUT KEY IN THE IGNITION SWITCH.TURN IGNITION TO THE ON POSITION.WHEN LED FLASHES TWICE THAT INDICATE KEY IS RECOGNIZED.IF KEY CODE NOT RECOGNIZED A MESSAGE ON THE TRIP COMPUTER WILL BE DISPLAYED.TO ENABLE ENGINE TO START WHEN KEY CODE NOT RECOGNIZED.SWITCH THE IGNITION ON.PRESS ANY KEY REMOTE TRANSMITTER BUTTON ONCE START THE ENGINE.IF ALL FAILS. THE KEY SWITCH TRANSPONDER FAILED OR THE KEY LOCK CYLINDER BROKEN, IMMOBILIZER MODULE FAULTY VECHICLE HAS TO BE SCANNED TO LOOK FOR FAULTY IMMOBILIZER MODULE.

That's your immobilizer system light:
The Immobilizer System protects your car from theft. A properly coded master or valet key must be used in the ignition switch for the engine to start.
If an improperly coded key (or other device) is used, the engine’s fuel system is disabled.
When you turn the ignition switch to ON (II), the Immobilizer System indicator should come on for a few seconds, then go out. If the indicator starts to blink, it means the system does not recognize the coding of the key. Turn the ignition switch to LOCK (0), remove the key, reinsert it, and turn the switch to ON (II) again.
The system may not recognize your key’s coding if another immobilizer key or other metal object is near the ignition switch when you insert the key. To make sure the system recognizes the key code:
Do not keep other immobilizer keys on the same key ring.
Use a plastic or leather key fob, not metal.
Keep other keys away from your vehicle’s key and the ignition switch while trying to start the engine.
Do not attempt to alter this system or add other devices to it. Electrical problems could result that may make your car undriveable.
This indicator will also blink several times when you turn the ignition switch from ON (II) to ACCESSORY (I) or LOCK (0).
If the system repeatedly does not recognize the coding of your key, contact your Honda dealer.
If you have lost your key and you cannot start the engine, contact your Honda dealer.
Hope this lets you understand the light, and what should be happening.

Dead Battery or factory alarm not recognizing the chip in your key.  Cheapest thing is to go get the spare key and see if that works.  Then you know its the key and that the alarm has to be reset to recognize the key.  Some cars want you to step on the brake before starting.