At least twice in the Revived Series, the Inspector has come face-to-face with places that exist outside of time.
One of those places turns out to be their home planet, Kayaclasch.

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At least twice in the Revived Series, the Inspector has come face-to-face with places that exist outside of time.
One of those places turns out to be their home planet, Kayaclasch.

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The Inspector has repeatedly said that the reason he departed from his home planet was
the extensive decadence and poor morals of his fellow Infinity Knights. Not to mention their gross xenophobia.
Again and again, the programme likes to circle back
to Kayaclasch, from where the Inspector fled at the very beginning.
Kayaclasch – Inspector Spacetime
Kayaclasch is the home planet of the Infinity Knights, including the Inspector and the Sergeant. The whole planet was apparently destroyed i
Interplanetary Rescue Patrol: Come in, unidentified spacecraft. What is your position? Repeat, what is your position? The Inspector: My position? Well, I'm a Detective Inspector of the Infinity Knights from the planet Kayaclasch in the Boötes Galaxy."
When it comes to pure scenery-chewing, *Inspector Spacetime* hit a high note
with the introduction of Omnicron, the alleged creator of Kayaclasch and the Infinity Knights. It’s amazing anything was left standing after that performance.
‘Kayaclasch was a dream that I, OMNICRON, made into being, through sheer FORCE OF WILL. I created you out of ... DUST ... and ... LIGHT! I ALONE MADE ALL THIS! I AM THE CREATOR!!!’

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The architecture and clothing of Kayaclasch is a bit of a mix of retro and futuristic.
Which also papers over the stagnation and decline of Infinity Knight society.
The Revived Series started off with a cataclysmic backstory,
the Time Wave having destroyed Kayaclasch and leaving the Inspector as the sole Infinity Knight. (Or so he thought.)
If Kayaclasch really is in a tertiary star system,
why do we only see, at most, two suns in the sky at any given moment?