Deferred, not yet a number
This is the chapter that trips people up, so we are going to meet it directly rather than let it ambush you later. In this world, most values — and nearly all the interesting ones — are not numbers. They are deferred operations: computations that have not collapsed to a numeral, and shouldn't. Once that lands, almost nothing in the rest of the book feels strange.
Take an ordinary value like 3. It is completely settled. There is nothing about "how it got here" that you need to keep, because you can do anything to it and always find your way back without that history. Reversibility asks nothing extra of it. A plain number is a value that has to remember nothing.
Now take √−1. Try to boil it down to a plain number on the line and you can't — there is no such number. But look at why, in the language of the last chapter. To be able to reverse it, the value has to carry three things: a number (1), a sign (the minus), and an operation (the square root). Throw any of those away and you have destroyed exactly the information you'd need to undo it — which is the one thing we've forbidden. So it doesn't collapse. It stays as what it is: a held, unfinished operation.
You might think the native notation would fix this — write it as 0^(ω/2) instead and surely it becomes "a value" rather than "an operation." It doesn't, and this is the point. 0^(ω/2) carries the same three pieces of information as √−1 — a base, a power, a direction. The notation changed; the amount of history that must be recorded did not. Whatever you call it, the value is exactly as deferred as reversibility requires it to be.
So here is the inversion, stated plainly.
Read the usual picture and this one side by side. The usual picture says numbers are the real things and operations act on them. This one says the operational history is the real thing, and a scalar is the degenerate case whose history is all shadow — nothing about it lives off the line, so a numeral loses none of it. The interesting values live partly off the line — carrying a little such history, or a lot — and they never finish resolving into a numeral, because finishing would discard the part that was never on the line to begin with.
The discomfort is real and worth naming: we are trained from childhood that every value must "come out to a number," and a value that stays an operation feels unfinished, like an answer someone forgot to compute. But that instinct is the very habit this theory rejects. Forcing every value down to a numeral is the information-destruction we outlawed in the last chapter — it is the same move as letting zero swallow what it multiplies. A deferred value is not a placeholder for some "real" answer waiting to be found. It is the answer, kept in the only form that doesn't throw part of itself away.
Everything ahead gets easier once you stop trying to collapse. The reciprocal of zero, the leftovers of a cancellation, the differential that falls out of multiplying by zero — each is a deferred operation carrying precisely the history it needs, no more. When one of them shows up looking like "not a real number," that is not a warning sign. It is the theory working exactly as defined.