Keys and Filters

Turn a key in a lock. It clicks. Turn it back. It unclicks. Nothing is lost. That is a unit.

Now pour coffee through a filter. The liquid passes; the grounds stay behind. You cannot un-filter coffee. That is a projection.

In the thin ring (2310), there are 480 keys and 32 filters. The True Form (970,200) has 201,600 keys but still only 32 filters — the same 5-cube, unchanged. Together they make 480 + 32 = 512 = 2⁹. Everything else? Broken keys. They turn, but something jams. Information is lost. And you can never get it back.

What Kind of Thing Are You?

Every number in the ring is one of three things: a key that opens and closes cleanly, a filter that sorts but can never unsort, or something broken that loses information along the way. Which kind is your number? Type one and find out.

Five Light Switches

Imagine a room with five light switches on the wall. Each one controls a prime ring. Flip any combination you like: 2⁵ = 32 possible settings. Each setting IS a filter — an idempotent that decides which channels pass through and which go dark. Click the circles below. Feel what each switch does.

The Prison Theorem

Here is the startling thing. The moment you turn off even one switch, every single key in the room breaks. Not most of them — all 480. No exceptions. Freedom is total or it is nothing. Isn't that remarkable? Verified exhaustively: 0 violations across all 31 non-trivial filters and all 480 keys. You cannot have partial reversibility in this ring.

The Spectral Duality

Of the 288 eigenvalue classes, the 32 filters can reach 258 of them (SOLVE). The remaining 30 belong only to units (ACT). 258 + 30 = 288. The ring splits its spectrum into what can be projected and what can only be traversed.

Three Ways to Touch

Think about three different ways of touching something. You can look at it (add) — that never damages. You can handle it (multiply) — sometimes things jam. Or you can decide about it (project) — and once you decide, there is no undeciding. Each way is less reversible than the last.

*except e=1 (identity, which projects to everything)

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The Interactive Atlas · Z/2310Z → Z/970200Z · Chapter 5 of 14