There’s a strange asymmetry sitting under almost everything we do with computers: some things are easy to check and hard to find. A completed Sudoku takes seconds to verify and hours to solve. A factorization takes one multiplication to confirm and, for a large enough number, more time than the universe has to discover. We lean on this gap constantly — every password, every signature, every proof-of-work — and yet the gap itself is rarely examined as a thing in its own right.
Preservation and Horizons is my attempt to examine it. Where my other book, The Wall at Two, stays close to one concrete problem — why a semiprime hides its factors — this one steps back and asks the more general question: what is difficulty? Not as an inconvenience, but as a structure. Where does it live? What does it protect? And why does it have horizons — boundaries past which no clever reframing seems to help?
The two ideas in the title
Preservation is the observation that difficulty doesn’t leak. When you transform a hard problem — change coordinates, re-encode it, look at it from a different angle — the hardness travels with it. It’s conserved, the way physicists talk about conserved quantities. You can move it around, concentrate it, dilute it across a bigger structure, but you can’t make it vanish by being clever about notation. The book spends a lot of time on why this is, and on what the failed attempts to “dissolve” hard problems have in common.
Horizons is the geometric half. A horizon is a boundary you can see up to but not past — and computation is full of them. Verification gives you a view right up to the edge: you can confirm an answer sits exactly where it claims to. But finding the answer requires crossing the horizon, and the crossing costs something that checking never does. The book develops this picture through a series of thought experiments about infinite sharpness and infinite circles — how a curve can be perfectly defined at every point and still hold a location that no finite process pins down.
What kind of book this is
It’s not a textbook and it’s not a survey of complexity theory. There are no complexity classes in the first half at all. It’s closer to an extended essay — the kind of book you read with a pencil, arguing back. The chapters build a vocabulary of pictures: walls, hinges, shadows, horizons. Each one is introduced through a concrete case before it gets used to say anything general.
The later chapters take the framework somewhere I didn’t expect when I started: toward verification as the fundamental act. We tend to think of solving as primary and checking as an afterthought. The book argues the reverse — that what a hard problem preserves is precisely the value of its verification, and that this is why difficulty is useful at all. Money, trust, identity, proof: all of them are ways of spending the gap between finding and checking.
Who it’s for
Readers who liked the concrete geometry of The Wall at Two and wanted the wider view. Readers who never touched that book but have wondered why “hard to compute, easy to verify” keeps showing up everywhere from cryptography to puzzles to blockchain. Anyone who suspects that difficulty isn’t just an engineering obstacle but a feature of how structure works.
As with everything I publish, I’ll be direct about the ground I’m standing on: I’m an independent researcher working outside any institution. The arguments here are mine, they’re laid out step by step, and where I’m speculating rather than proving, the text says so plainly.
Where to get it
Preservation and Horizons: On Difficulty, Verification, and Structural Limits is available now in Kindle ($9.99) and paperback ($10.99) editions.