Beige Papers · a ground-up introduction

Constructive Operational Type Theory

One refusal — never destroy information — and everything it forces, built one step at a time.

DoozerDiffuser · sibarum.github.io/cott

This is an introduction to a small idea followed a long way. Insist that no operation ever throw information away, and a surprising amount of mathematics reorganizes itself: zero gets a reciprocal, the imaginary unit turns out to be hiding inside it, and the circle can be drawn in whole-number arithmetic with no limits at all. You can read straight through, each chapter forced by the one before it, or follow the links sideways wherever your curiosity pulls. No mathematics beyond high-school algebra is assumed.

How claims are marked

Not everything here stands on the same ground, and the difference is marked honestly rather than hidden. Most of the book is plain reasoning that needs no badge. Where a specific claim has a particular standing, it wears one of three:

Proven   a genuine theorem, or a result checked exactly by machine.
Plausible   a load-bearing bet — stated as one, not yet proven.
Wild   where the thread points past its proof; read as speculation.
Contents
The One Idea
  1. 1Types as OperationsWhat a number is for, not what it is
  2. 2Totality and ReversibilityThe one thing we refuse to do
  3. 3What a Value IsDeferred, not yet a number
  4. 4The MethodDelete one axiom, keep everything else
  5. 5Cancellation as an OperationZero and one are what's left behind
  6. 6The Invertible ZeroA reciprocal for nothing — and why it isn't infinity
The Circle for Free
  1. 7Powers of the ZeroWhere −1 comes from
  2. 8The Imaginary UnitA zero seen edge-on
  3. 9The Exact CirclePhase without limits; roots of unity by hand
The Two Operations
  1. 10Multiplication, SolvedBookkeeping with grades
  2. 11The Addition ProblemThe wound the grades can't close
  3. 12The Addition DialEvery average is a different plus
  4. †Where the Choices ShowBeyond the dial: the deeper result that fixes 0ω
Calculus Without Limits
  1. 13The Structural DifferentialDifferentiation as multiplication by zero
  2. 14The ChartOne derivative wearing three hats
  3. 15Void Calculus in PracticeSlopes and roots without a limit
  4. 16The Four Registers soonWhere the method stops and analysis begins
Obstructions
  1. 17Phases That Won't Close soonThe residue as a winding number
  2. 18The Octonion Associator soonA sign that survives every disguise
  3. 19One Object, Three Faces? soonA conjecture about residues, associators, and moduli
The Frontier
  1. 20Reversible vs One-Way soonKeep the receipt, or burn it
  2. 21Order, Time, and Physics soonPast the proof
Judging the Work
  1. 22Honest Logic soonReceipts, and what it takes to call something true
  2. 23The Debt Ledger soonThe two things everything rests on
Appendix
  1. AThe Proven ResultsThe three theorems the book leans on, stated and proved
Published under the CC BY 4.0 license. Corrections and counterexamples welcome — in this framework a good counterexample is information, not defeat.